Selecting Procedures for Calculating Derivatives - AP Calculus AB

For $G(x)=\dfrac{e^{x}}{\sqrt{x^2+9}}$ , which differentiation rules should be used to compute $G'(x)$ ?
Answer: Quotient rule and chain rule This problem requires selecting differentiation procedures for $G(x)=\dfrac{e^{x}}{\sqrt{x^2+9}}$ . Since this is a quotient, the quotient rule is necessary. Additionally, the denominator $\sqrt{x^2+9}$ is a composite function where the outer function is the square root and the inner function is $x^2+9$ , requiring the chain rule when differentiating the denominator. The numerator $e^x$ differentiates to itself. The product rule alone wouldn't work since this is a quotient structure, and implicit differentiation isn't needed for this explicit function. When you have a quotient where either part contains a composite function, combine the quotient rule with the chain rule.
For $k(x)=\sqrt{x^2+\cos x}$ , which differentiation rules are needed to compute $k'(x)$ ?
Answer: Chain rule Selecting the appropriate differentiation procedure requires recognizing composite function structures. The function $k(x)=\sqrt{x^2+\cos x}$ can be rewritten as $(x^2+\cos x)^{1/2}$ , which is clearly a composite function. The chain rule is the necessary and sufficient tool here: we differentiate the outer function (square root or power of 1/2) and multiply by the derivative of the inner function $(x^2+\cos x)$ . The product rule is not needed because we don't have multiplication of separate functions, and the quotient rule is irrelevant as there's no division. While we use basic differentiation rules for $x^2$ and $\cos x$ within the chain rule application, the chain rule is the primary procedure. The recognition strategy is to identify when one function is "inside" another—square roots, powers, and other function compositions all signal the need for the chain rule.
For $u(x)=\sqrt{1-\sin x}$ , which differentiation rules are required to find $u'(x)$ ?
Answer: Chain rule applied twice This problem requires selecting differentiation procedures for $u(x)=\sqrt{1-\sin x}$ . This function involves nested composition: the outer function is the square root, and the inner function is $1-\sin x$ . The chain rule must be applied twice - first for the square root, then for the sine function within the expression $1-\sin x$ . Working from outside to inside: the derivative of $\sqrt{u}$ is $\frac{1}{2\sqrt{u}}$ , and then we need the derivative of $1-\sin x$ , which is $-\cos x$ . The product rule doesn't apply since this is a single composite function, and implicit differentiation isn't needed. When you encounter functions with multiple layers of composition, apply the chain rule repeatedly, working from the outermost function inward.
Let $y$ be defined by $e^{y}+xy=4$ . Which differentiation procedure is needed to find $\dfrac{dy}{dx}$ ?
Answer: Implicit differentiation (including product rule on $xy$ ) This problem involves selecting the differentiation procedure for finding $\dfrac{dy}{dx}$ when $y$ is defined implicitly by $e^{y}+xy=4$ . Since $y$ cannot be easily isolated as an explicit function of $x$ , implicit differentiation is required. This involves differentiating both sides with respect to $x$ , treating $y$ as a function of $x$ . The term $e^y$ requires the chain rule (giving $e^y \frac{dy}{dx}$ ), and the term $xy$ requires the product rule (giving $y + x\frac{dy}{dx}$ ). Neither quotient rule alone nor explicit function methods apply to this implicit relationship. When you have an equation mixing $x$ and $y$ terms where $y$ cannot be isolated, implicit differentiation with appropriate subsidiary rules is necessary.
For $p(x)=x^4\,e^{x^2}$ , which differentiation rule(s) should be used to find $p'(x)$ ?
Answer: Product rule and chain rule Selecting the appropriate differentiation procedure requires recognizing both product and composite structures. The function $p(x)=x^4 e^{x^2}$ is the product of two functions: $x^4$ and $e^{x^2}$ . The product rule tells us to differentiate as $(x^4)' \cdot e^{x^2} + x^4 \cdot (e^{x^2})'$ . The first term gives us $4x^3 e^{x^2}$ using the power rule. For the second term, we need the chain rule to differentiate $e^{x^2}$ because $x^2$ is inside the exponential function, giving us $e^{x^2} \cdot 2x$ . Using only the chain rule would miss the product structure, while using only the product rule would incorrectly handle the composite exponential. The recognition strategy is to first identify the main operation (here, multiplication), apply the appropriate rule (product rule), then examine each component for additional complexity (like the composite exponential requiring chain rule).
A function is $E(x)=e^{\sin(3x)}$ . Which differentiation rule(s) are required to find $E'(x)$ ?
Answer: Chain rule applied multiple times This problem requires selecting differentiation procedures for $E(x)=e^{\sin(3x)}$ . This function involves multiple layers of composition: the outer function is the exponential, the middle function is the sine, and the inner function is $3x$ . The chain rule must be applied multiple times to handle this nested structure. Starting from the outside: the derivative of $e^u$ is $e^u \frac{du}{dx}$ , then we need the derivative of $\sin(3x)$ , which requires another application of the chain rule giving $\cos(3x) \cdot 3$ . Neither product rule, quotient rule, nor implicit differentiation apply since this is a single composite function. When you see exponential functions with composite arguments that are themselves composite, apply the chain rule repeatedly.
For $r(x)=\dfrac{\tan x}{x^2+3x}$ , which differentiation rules are needed to compute $r'(x)$ ?
Answer: Quotient rule Selecting the appropriate differentiation procedure requires careful analysis of the function's structure. The function $r(x)=\frac{\tan x}{x^2+3x}$ is a quotient with $\tan x$ in the numerator and a polynomial in the denominator. The quotient rule is the primary tool needed here because we have one function divided by another. While differentiating $\tan x$ technically involves knowing that $\frac{d}{dx}[\tan x] = \sec^2 x$ , this is a standard derivative that doesn't require applying the chain rule in this context. The denominator $x^2+3x$ is differentiated using basic power rule. Neither product rule nor chain rule is needed for this particular function. The recognition strategy is straightforward: when you see a fraction bar with functions above and below, the quotient rule is your primary tool.
For $R(x)=\dfrac{\cos(3x)}{(x^2+1)^2}$ , which differentiation rules should be used to find $R'(x)$ ?
Answer: Quotient rule and chain rule This problem involves selecting differentiation procedures for $R(x)=\dfrac{\cos(3x)}{(x^2+1)^2}$ . Since this is a quotient, the quotient rule is necessary. Additionally, the numerator $\cos(3x)$ is a composite function requiring the chain rule (outer function is cosine, inner function is $3x$ ), and the denominator $(x^2+1)^2$ is also composite, requiring the chain rule (outer function is square, inner function is $x^2+1$ ). This demonstrates how the quotient rule often combines with the chain rule when dealing with composite functions in both numerator and denominator. When you have a quotient where both parts contain composite functions, apply the quotient rule with the chain rule for each part.
For $d(x)=\dfrac{\ln(x)}{x^2+1}$ , which differentiation rule(s) should be used to compute $d'(x)$ ?
Answer: Quotient rule This problem requires selecting the appropriate differentiation procedure for $d(x)=\dfrac{\ln(x)}{x^2+1}$ . Since this function is a quotient with both numerator and denominator being functions of $x$ , the quotient rule is the correct approach. The numerator $\ln(x)$ differentiates to $\frac{1}{x}$ , and the denominator $x^2+1$ differentiates to $2x$ using basic differentiation rules. The product rule doesn't apply since this is a quotient structure, and chain rule alone isn't sufficient since we need to handle the division properly. When you encounter a fraction where both numerator and denominator are functions of the variable, the quotient rule is the standard method.
For $y$ defined implicitly by $x^2y+\cos(y)=3$ , which differentiation method should be used to find $\dfrac{dy}{dx}$ ?
Answer: Implicit differentiation, using product rule and chain rule This problem involves finding the derivative when $y$ is defined implicitly by an equation. The equation $x^2y+\cos(y)=3$ cannot be easily solved for $y$ in terms of $x$ , making implicit differentiation the appropriate method. When differentiating implicitly, we treat $y$ as a function of $x$ and apply the chain rule whenever we differentiate terms containing $y$ . The term $x^2y$ requires the product rule (since it's a product of $x^2$ and $y$ ), and $\cos(y)$ requires the chain rule (since $y$ is a function of $x$ ). Logarithmic differentiation would be inappropriate here, and trying to use only the product rule would miss the implicit relationship. When variables are mixed in an equation that defines their relationship, implicit differentiation is the key procedure.
Given $n(x)=\sec(x^3-2x)$ , which differentiation rule is needed to find $n'(x)$ ?
Answer: Chain rule (with the derivative of $\sec$ ) This problem involves selecting the appropriate differentiation procedure for $n(x)=\sec(x^3-2x)$ . This function is a composition where the outer function is the secant function and the inner function is $x^3-2x$ . The chain rule is required, using the derivative formula for secant: $\frac{d}{dx}[\sec(u)] = \sec(u)\tan(u) \cdot \frac{du}{dx}$ . The inner function $x^3-2x$ differentiates to $3x^2-2$ . The product rule and quotient rule don't apply since this is a single composite function, and implicit differentiation isn't needed. When you encounter trigonometric functions with composite arguments, use the chain rule with the appropriate trigonometric derivative formula.
A population is modeled by $P(t)=(2t-1)^6$ . Which differentiation rule is sufficient to find $\dfrac{dP}{dt}$ ?
Answer: Chain rule Identifying the minimal sufficient differentiation procedure is a key skill in calculus. The population model $P(t)=(2t-1)^6$ is a composite function where a linear expression $(2t-1)$ is raised to the sixth power. The chain rule alone is sufficient here: we differentiate the outer function (something to the sixth power) and multiply by the derivative of the inner function $(2t-1)$ . The product rule is unnecessary because we don't have multiplication of separate functions, and the quotient rule is irrelevant as there's no division. While we do use the power rule as part of applying the chain rule, the chain rule is the primary procedure needed. To recognize such cases, look for a single composite function structure—when one function is "inside" another, the chain rule suffices.
Let $D(x)=\sqrt[3]{(2x-5)^2}$ . Which differentiation rule(s) should be used to find $D'(x)$ ?
Answer: Chain rule (rewrite as a rational power) This problem involves selecting the appropriate differentiation procedure for $D(x)=\sqrt[3]{(2x-5)^2}$ . This can be rewritten as $D(x)=[(2x-5)^2]^{1/3} = (2x-5)^{2/3}$ , which is a composition where the outer function is the power $2/3$ and the inner function is $2x-5$ . The chain rule is required, treating this as a rational power of a linear function. The quotient rule isn't needed since this can be handled as a single power, and logarithmic differentiation would be unnecessarily complex. When you encounter radical expressions that can be rewritten using rational exponents, the chain rule with power rule is typically the most efficient approach.
A function is $W(x)=\sqrt{x^2+1}+\dfrac{3}{x^2}$ . Which differentiation rules are needed to compute $W'(x)$ ?
Answer: Chain rule and power rule (with sum rule) This problem involves selecting differentiation procedures for $W(x)=\sqrt{x^2+1}+\dfrac{3}{x^2}$ . This function is a sum of two terms, so the sum rule applies. The first term $\sqrt{x^2+1}$ requires the chain rule (outer function is square root, inner function is $x^2+1$ ), and the second term $\dfrac{3}{x^2} = 3x^{-2}$ requires the power rule. Neither product rule, quotient rule at the top level, nor implicit differentiation apply since we're dealing with a sum where each term can be differentiated using basic rules. When you encounter sums of functions where individual terms may be composite, apply the sum rule and use appropriate rules for each term.
For $h(x)=\ln\left(\dfrac{(x^2+1)^5}{\sqrt{3x-2}}\right)$ , which rules should be used to find $h'(x)$ ?
Answer: Chain rule and properties of logarithms to simplify first Selecting the appropriate differentiation procedure often involves recognizing when simplification can make the process more efficient. The function $h(x)=\ln\left(\frac{(x^2+1)^5}{\sqrt{3x-2}}\right)$ appears complex, but logarithm properties allow us to simplify before differentiating. Using the properties $\ln(a/b) = \ln(a) - \ln(b)$ and $\ln(a^n) = n\ln(a)$ , we can rewrite this as $h(x) = 5\ln(x^2+1) - \frac{1}{2}\ln(3x-2)$ . Now we can differentiate using the chain rule on each logarithmic term, which is much simpler than applying the quotient rule to the original expression. While the quotient rule could technically work on the argument inside the logarithm, it would create unnecessary complexity. The recognition strategy is to always check if logarithm properties can simplify your expression before differentiating—this often transforms a complex quotient or product into a sum or difference that's easier to handle.
For $z(x)=\dfrac{\sqrt{x}}{\ln x}$ , which differentiation rule(s) should be used to compute $z'(x)$ ?
Answer: Quotient rule This problem involves selecting the appropriate differentiation procedure for $z(x)=\dfrac{\sqrt{x}}{\ln x}$ . Since this function is a quotient with both numerator and denominator being functions of $x$ , the quotient rule is the correct approach. The numerator $\sqrt{x} = x^{1/2}$ differentiates using the power rule to give $\frac{1}{2}x^{-1/2}$ , and the denominator $\ln x$ differentiates to $\frac{1}{x}$ . Neither product rule, chain rule alone, nor implicit differentiation apply to this quotient structure. When you encounter a fraction where both numerator and denominator are functions of the variable (even if they're basic functions), the quotient rule is the standard method.
Let $y$ be defined by $\ln(y)=x^2-3x$ . Which differentiation procedure is required to find $\dfrac{dy}{dx}$ ?
Answer: Implicit differentiation This problem requires selecting the differentiation procedure for finding $\dfrac{dy}{dx}$ when $y$ is defined implicitly by $\ln(y)=x^2-3x$ . Since $y$ appears inside a logarithm and cannot be easily isolated as an explicit function of $x$ , implicit differentiation is required. Differentiating both sides with respect to $x$ : the left side gives $\frac{1}{y} \frac{dy}{dx}$ using the chain rule, and the right side gives $2x-3$ . The equation can then be solved for $\frac{dy}{dx}$ . Neither explicit function rules nor other standard rules apply directly to this implicit relationship. When you have an equation where the dependent variable appears inside functions and cannot be isolated, implicit differentiation is the necessary approach.
A population is modeled by $P(t)=\dfrac{t^2+3t}{\ln(t+1)}$ . Which differentiation rule(s) should be used to compute $P'(t)$ ?
Answer: Quotient rule and chain rule This problem requires selecting appropriate differentiation procedures for $P(t)=\dfrac{t^2+3t}{\ln(t+1)}$ . Since this is a fraction with both numerator and denominator being functions of $t$ , the quotient rule is necessary. Additionally, the denominator $\ln(t+1)$ is a composite function requiring the chain rule to differentiate properly. The product rule alone wouldn't work since this is a quotient, and the power rule is insufficient for handling the logarithmic denominator. Logarithmic differentiation is not required here as we can directly apply the quotient rule. When you see a fraction where the denominator contains a composite function, expect to use both quotient rule and chain rule.
Given $U(x)=x^2\,e^{\sqrt{x}}$ , which differentiation rules are needed to compute $U'(x)$ ?
Answer: Product rule and chain rule This problem involves selecting differentiation procedures for $U(x)=x^2\,e^{\sqrt{x}}$ . This function is a product of two functions: $x^2$ and $e^{\sqrt{x}}$ , so the product rule is necessary. Additionally, the second factor $e^{\sqrt{x}}$ is a composite function where the outer function is the exponential and the inner function is $\sqrt{x}$ , requiring the chain rule. The first factor $x^2$ differentiates to $2x$ using the power rule. This demonstrates how the product rule combines with the chain rule when one factor involves nested composition. When you encounter a product where one factor is a composite exponential function, expect to use both product rule and chain rule.
For $q(x)=\left(\dfrac{x^2-1}{x^2+1}\right)^6$ , which differentiation rule(s) should be used to find $q'(x)$ ?
Answer: Chain rule and quotient rule Identifying the correct differentiation procedure requires recognizing a power of a quotient, which creates a composite function. The function $q(x)=\left(\frac{x^2-1}{x^2+1}\right)^6$ has the structure of an inner function (the quotient $\frac{x^2-1}{x^2+1}$ ) raised to the 6th power, requiring the chain rule. We first apply the power rule to get $6\left(\frac{x^2-1}{x^2+1}\right)^5$ , then multiply by the derivative of the inner quotient. To find the derivative of $\frac{x^2-1}{x^2+1}$ , we need the quotient rule. Using only the quotient rule would ignore the outer power structure, while using only the power rule would incorrectly treat the quotient. The key insight is recognizing nested operations: when you see an entire expression (especially a fraction) raised to a power, you'll need the chain rule for the power and additional rules for whatever's inside the parentheses.

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