Selecting Procedures for Calculating Derivatives - AP Calculus BC

1. Given $s(x)=\dfrac{\sin(2x)}{x^3-1}$, which differentiation technique is most appropriate to find $s'(x)$?​
   Answer: Use the quotient rule with the chain rule This problem requires choosing the best approach for a quotient involving a composite function in the numerator. The function $s(x)=\frac{\sin(2x)}{x^3-1}$ is a quotient where the numerator $\sin(2x)$ is itself a composite function. The quotient rule is necessary for the overall structure, and when differentiating $\sin(2x)$, the chain rule must be applied. The quotient rule gives $\frac{(x^3-1)\cdot\frac{d}{dx}[\sin(2x)] - \sin(2x)\cdot\frac{d}{dx}[x^3-1]}{(x^3-1)^2}$, where $\frac{d}{dx}[\sin(2x)] = 2\cos(2x)$ by the chain rule. Using implicit differentiation (option C) adds unnecessary complexity to an explicitly defined function. For quotients where numerator or denominator involves composition, combine the quotient rule with the chain rule as needed.
2. A model uses $H(x)=\big(\sqrt{x^2+9}\big)^5$. Which differentiation technique is most appropriate to find $H'(x)$?
   Answer: Differentiate using the chain rule Selecting the appropriate differentiation technique is a key skill in calculus, ensuring efficiency and accuracy in finding derivatives. For H(x) = (√(x² + 9))^5, the chain rule is most appropriate because it's a composition: the outer power of 5 applied to the inner square root of (x² + 9). Rewriting as $(x² + 9)^{5/2}$ allows efficient differentiation using the power rule externally and chain rule for the inside, yielding H'(x) = (5/2)$(x² + 9)^{3/2}$ * 2x. This avoids expansion and keeps the form compact. Differentiating by expanding into $x^5$ + $9^5$ is incorrect because that's not the binomial expansion of the root. For powers of composite functions, prioritize the chain rule to preserve the original structure and reduce algebraic burden.
3. If $r(x)=\sqrt{1+\tan(3x)}$, which differentiation technique is most appropriate to compute $r'(x)$?​
   Answer: Use the chain rule repeatedly This problem involves selecting the technique for a nested composition of functions. The function $r(x)=\sqrt{1+\tan(3x)}$ can be written as $(1+\tan(3x))^{1/2}$, revealing a chain of compositions: the square root of (1 plus the tangent of (3 times x)). The chain rule must be applied repeatedly: first for the square root, then for the tangent function, and finally for the linear function $3x$. This gives $\frac{1}{2}(1+\tan(3x))^{-1/2} \cdot \sec^2(3x) \cdot 3$. Using logarithmic differentiation (option D) would work but is unnecessarily complex for this straightforward composition. When functions are nested compositions, identify each layer and apply the chain rule systematically from outside to inside.
4. For $f(x)=\sqrt{5x^3-2x+7}$, which differentiation technique is most appropriate to find $f'(x)$ efficiently?
   Answer: Differentiate using the chain rule by rewriting as $(5x^3-2x+7)^{1/2}$ This problem asks for the most efficient technique to differentiate a square root of a polynomial. The function $f(x)=sqrt{5x^3-2x+7}$ can be rewritten as $(5x^3-2x+7)^{1/2}$, revealing it as a composite function perfect for the chain rule. Applying the chain rule gives $f'(x) = \frac{1}{2}(5x^3-2x+7)^{-1/2} cdot (15x^2-2) = \frac{15x^2-2}{2sqrt{5x^3-2x+7}}$. Option E's suggestion to split the square root as a product is mathematically incorrect since $sqrt{a+b} \neq sqrt{a}cdotsqrt{b}$. When differentiating roots or fractional powers of expressions, rewriting with rational exponents and using the chain rule is the standard efficient approach.
5. A model is $q(x)=(x^2+1)^{3}(2x-5)^{4}$. Which differentiation technique is most appropriate to find $q'(x)$?
   Answer: Differentiate using logarithmic differentiation to turn products and powers into sums This problem requires selecting the technique for a product of two functions raised to powers. The function $q(x)=(x^2+1)^3(2x-5)^4$ is a product of powers, making logarithmic differentiation particularly efficient. Taking the natural log: $ln(q) = 3ln(x^2+1) + 4ln(2x-5)$. Differentiating implicitly: $\frac{q'}{q} = \frac{3(2x)}{x^2+1} + \frac{4(2)}{2x-5}$, so q'(x) = (x^2+1)^3(2x-5)^4left[\frac{6x}{x^2+1} + \frac{8}{2x-5}\right]. Option B's product rule approach would require multiple applications and be much more tedious. When differentiating products of multiple factors raised to powers, logarithmic differentiation converts the problem into a simpler sum, making it the most efficient technique.
6. A revenue function is $R(x)=x\ln(4x+1)$. Which differentiation technique is most appropriate to find $R'(x)$?
   Answer: Differentiate using the product rule with a chain rule on $\ln(4x+1)$ This problem requires selecting the appropriate technique for differentiating a product of functions. The function $R(x)=xln(4x+1)$ is a product of $x$ and $ln(4x+1)$, so the product rule is necessary. Additionally, $ln(4x+1)$ is a composite function requiring the chain rule. Using the product rule: $R'(x) = 1 cdot ln(4x+1) + x cdot \frac{1}{4x+1} cdot 4 = ln(4x+1) + \frac{4x}{4x+1}$. Option A's suggestion to use only the chain rule ignores the product structure entirely. When you have a product where at least one factor is composite, combine the product rule with the chain rule as needed.
7. For $p(x)=\big(\sin x\big)^5\big(\cos x\big)^3$, which differentiation technique is most appropriate to find $p'(x)$?​
   Answer: Use the product rule combined with the chain rule This problem involves selecting the appropriate technique for a product of powers of trigonometric functions. The function $p(x)=(\sin x)^5(\cos x)^3$ is a product of two factors, each involving a composition (power of a trig function). The product rule is necessary to handle the multiplication, and within each term of the product rule, the chain rule is needed for the powers of sine and cosine. Applying the product rule gives $(\sin x)^5 \cdot \frac{d}{dx}[(\cos x)^3] + (\cos x)^3 \cdot \frac{d}{dx}[(\sin x)^5]$, where each derivative requires the chain rule. Using implicit differentiation only (option C) would be unnecessarily complex for this explicit function. When differentiating products where each factor is itself a composite function, combine the product rule with the chain rule for efficient computation.
8. Let $y=x^{\sin x}$. Which differentiation technique is most appropriate to find $\dfrac{dy}{dx}$?
   Answer: Differentiate using logarithmic differentiation by taking $\u007fln\u007f$ of both sides. Selecting the appropriate differentiation technique is essential for functions with variable bases and exponents in calculus. Logarithmic differentiation is most efficient by taking the natural log of both sides, yielding ln y = sin x * ln x, and then differentiating implicitly. This allows handling the variable exponent easily, resulting in y' = y (cos x ln x + sin x / x). It avoids difficulties with direct power rule application since the exponent is not constant. A tempting distractor is using the power rule treating sin x as constant, but this fails because the exponent varies with x, making the standard power rule inapplicable. For exponential expressions with variables in both base and exponent, always use logarithmic differentiation to simplify the process effectively.
9. An electrical signal is modeled by $V(t)=\ln\big((t^2+4)^3\big)$. Which technique is most appropriate to find $V'(t)$?
   Answer: Differentiate using the chain rule with properties of logarithms Selecting the appropriate differentiation technique is a key skill in calculus, ensuring efficiency and accuracy in finding derivatives. For V(t) = ln($(t² + 4)^3$), using the chain rule with properties of logarithms is most appropriate because simplifying to 3 ln(t² + 4) first allows easy differentiation with the chain rule on the inner function. This approach leverages the logarithm power rule to reduce complexity before applying the derivative. It results in V'(t) = 3 (1/(t² + 4)) 2t, which is clean and direct. Differentiating using the product rule would be misguided since the function isn't a product after simplification. Consider logarithmic properties to simplify expressions before differentiation, especially with exponents inside logs, for a more streamlined process.
10. For $m(x)=\ln\!\left(\sqrt{1+x^2}\right)$, which differentiation technique is most appropriate to compute $m'(x)$?
    Answer: Differentiate using log properties to rewrite as $\tfrac12\ln(1+x^2)$, then apply the chain rule. Selecting the appropriate differentiation technique is vital for simplifying logarithmic expressions involving roots in calculus. Using log properties to rewrite as (1/2) ln(1 + x²) allows for easy differentiation with the chain rule, yielding (1/2) (2x) / (1 + x²) = x / (1 + x²). This simplification reduces the function to a scalar multiple of a basic log, making the derivative immediate. It avoids dealing with the square root directly in the differentiation process. A tempting distractor is using the product rule after rewriting as ln(√(1 + x²)) 1, but this doesn't simplify and still requires a chain rule on the log without the benefit of property reduction. Always apply logarithm properties to simplify arguments, especially with powers or roots, before selecting differentiation techniques.
11. In a mixing model, $R(x)=\dfrac{x^2+1}{\sqrt{3x-2}}$. Which differentiation technique is most appropriate for $R'(x)$?
    Answer: Differentiate using the quotient rule Selecting the appropriate differentiation technique is a key skill in calculus, ensuring efficiency and accuracy in finding derivatives. For R(x) = (x² + 1)/√(3x - 2), the quotient rule is most appropriate because the function is a ratio of two functions, with the denominator involving a composition that requires the chain rule internally. This method efficiently handles the structure by differentiating the numerator and denominator separately while applying the chain rule to the square root. It yields a precise derivative without needing to rewrite the function drastically. Differentiating by expanding into a polynomial first is not feasible since the square root prevents polynomial expansion. Evaluate the overall form of the function to choose between quotient, product, or chain rules for optimal efficiency.
12. For $f(x)=(3x^2-5x+1)^7$, which differentiation technique is most appropriate to find $f'(x)$ efficiently?​
    Answer: Use the chain rule with the power rule This problem requires selecting the most efficient technique for differentiating a composite function. The function $f(x)=(3x^2-5x+1)^7$ has the form $(u(x))^n$ where $u(x)=3x^2-5x+1$ is the inner function and the outer function raises it to the 7th power. The chain rule with the power rule is ideal here: differentiate the outer function (getting $7(3x^2-5x+1)^6$) and multiply by the derivative of the inner function (getting $6x-5$). While expanding the polynomial completely (option B) would work, it would require expanding a 7th power polynomial and differentiating many terms, making it highly inefficient. The key strategy is to recognize composite functions of the form $(u(x))^n$ and apply the chain rule rather than expanding.
13. A particle's position is $s(t)=t^3\big(2t^2-1\big)$. Which technique is most appropriate to compute $s'(t)$ efficiently?
    Answer: Differentiate using the product rule Selecting the appropriate differentiation technique is a key skill in calculus, ensuring efficiency and accuracy in finding derivatives. For s(t) = t³(2t² - 1), the product rule is most appropriate because the function is a clear product of two differentiable functions, t³ and (2t² - 1). Applying the product rule gives s'(t) = 3t²(2t² - 1) + t³(4t), which simplifies efficiently to 3t²(2t² - 1) + $4t^4$. This approach is direct and avoids unnecessary complications from other methods. Using the quotient rule would be incorrect and inefficient here since the function is not expressed as a quotient. Always identify if a function is a product, quotient, or composition to select the rule that matches its form and promotes simplicity.
14. A height function is $h(x)=\sqrt{(x^2+1)^3}$; which differentiation technique is most appropriate for $h'(x)$?
    Answer: Differentiate using the chain rule after rewriting as $(x^2+1)^{3/2}$. Selecting the appropriate differentiation technique is vital for efficiently handling nested powers and roots in calculus. Rewriting the function as $(x² + 1)^{3/2}$ enables the direct application of the chain rule, treating it as a power of a composite function. This approach simplifies the derivative to (3/2)$(x² + 1)^{1/2}$ 2x, which is quick to compute. It preserves the compact form and avoids expanding or unnecessary manipulations. A tempting distractor is using the product rule after rewriting as √(x² + 1) (x² + 1), but this introduces an extra rule application without simplifying the process further, as it still requires a chain rule on the square root. For functions with exponents and roots, prioritize consolidating into a single power and using the chain rule to streamline differentiation.
15. For $t(x)=\big(x^2+4\big)\,e^{3x}$, which differentiation technique is most appropriate to compute $t'(x)$?​
    Answer: Use the product rule This problem involves selecting the technique for a product of an algebraic and an exponential function. The function $t(x)=(x^2+4)e^{3x}$ is a product of two factors: the polynomial $(x^2+4)$ and the exponential $e^{3x}$. The product rule is the appropriate choice, giving $(x^2+4)\cdot\frac{d}{dx}[e^{3x}] + e^{3x}\cdot\frac{d}{dx}[x^2+4]$. Note that differentiating $e^{3x}$ requires the chain rule, yielding $3e^{3x}$. Using the chain rule only (option A) would be incorrect because this is a product, not a composition. When differentiating products of different function types (polynomial, exponential, trigonometric), the product rule is the primary tool, with the chain rule applied as needed within each factor.
16. For $p(x)=(x^2-1)\sin(4x)$, which technique is most appropriate to compute $p'(x)$?
    Answer: Differentiate using the product rule, with a chain rule on $\sin(4x)$. Selecting the appropriate differentiation technique is important for products involving polynomials and trigonometric functions in calculus. The product rule is most efficient here, applying it to (x² - 1) and sin(4x), with a chain rule on the sine term to account for the inner 4x. This yields 2x sin(4x) + (x² - 1) * 4 cos(4x), which is direct and requires minimal rewriting. It leverages the product structure naturally without complicating the expression. A tempting distractor is rewriting as a quotient using csc(4x), but this unnecessarily introduces the quotient rule and trigonometric reciprocals, making the differentiation more cumbersome. When faced with products, opt for the product rule combined with chain rules as needed to maintain simplicity and efficiency.
17. For $h(x)=e^{x^3-5x}\,(x^2+1)$, which differentiation technique is most appropriate to find $h'(x)$?
    Answer: Differentiate using the product rule with a chain rule on $e^{x^3-5x}$ This problem asks for the technique to differentiate a product involving an exponential function. The function $h(x)=e^{x^3-5x}(x^2+1)$ is a product of $e^{x^3-5x}$ and $(x^2+1)$, requiring the product rule. Additionally, $e^{x^3-5x}$ is composite, needing the chain rule. Using both: $h'(x) = e^{x^3-5x}(3x^2-5)(x^2+1) + e^{x^3-5x}(2x) = e^{x^3-5x}[(3x^2-5)(x^2+1) + 2x]$. Option B's suggestion to expand $e^{x^3-5x}$ is impossible since exponential functions don't expand into polynomials. When differentiating products involving composite exponential functions, combine the product rule with the chain rule.
18. For $g(x)=\ln\!\left(\dfrac{x^2+4}{x-1}\right)$, which technique is most appropriate to find $g'(x)$?
    Answer: Differentiate using log properties to rewrite as $\ln(x^2+4)-\ln(x-1)$, then apply derivative rules. Selecting the appropriate differentiation technique is crucial for simplifying logarithmic functions before differentiation in calculus. Using logarithm properties to rewrite the expression as $\ln(x^2 + 4) - \ln(x - 1)$ allows for straightforward differentiation of each term separately using the chain rule. This method breaks down the quotient inside the log into a difference, making the derivative computation simple and avoiding nested rules. It efficiently yields the result as $\frac{2x}{x^2 + 4} - \frac{1}{x - 1}$, which can be combined if needed. A tempting distractor is applying the quotient rule directly inside the logarithm without rewriting, but this complicates the process by requiring a chain rule on the entire log function combined with the quotient, leading to more steps. When dealing with logs of quotients, always apply log properties first to simplify and select techniques that reduce complexity.
19. For $m(x)=\arctan(3x^2-1)$, which differentiation technique is most appropriate to find $m'(x)$?
    Answer: Differentiate using the chain rule with the derivative of $\arctan(u)$ This problem asks for the technique to differentiate an inverse trigonometric function. The function $m(x)=arctan(3x^2-1)$ has arctangent as the outer function and $3x^2-1$ as the inner function, making the chain rule the appropriate choice. Using the chain rule with the derivative of arctangent: $m'(x) = \frac{1}{1+(3x^2-1)^2} cdot 6x = \frac{6x}{1+(3x^2-1)^2}$. Option B's suggestion to write arctangent as a product is mathematically nonsensical since $arctan(a+b) eq arctan(a)cdotarctan(b)$. When differentiating inverse trigonometric functions of expressions, the chain rule with the appropriate inverse trig derivative formula is the standard approach.
20. A position function is $s(t)=\sin(t^2+4t)$. Which differentiation technique is most appropriate to find $s'(t)$?
    Answer: Differentiate using the chain rule with outer sine and inner quadratic This problem requires selecting the technique for a composite trigonometric function. The function $s(t)=sin(t^2+4t)$ has sine as the outer function and $t^2+4t$ as the inner function, making it a perfect candidate for the chain rule. Using the chain rule: $s'(t) = cos(t^2+4t) cdot (2t+4) = (2t+4)cos(t^2+4t)$. Option A incorrectly suggests treating this as a product of $sin(t^2)$ and $sin(4t)$, which is not what the original function represents. When you have a trigonometric function of a polynomial expression, the chain rule is the standard and most efficient approach.

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