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.

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.

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.

This problem requires selecting the most efficient technique to differentiate a composite function. The function $P(t)=(3t^2-1)^5$ has an outer power function (raising to the 5th power) and an inner polynomial function $(3t^2-1)$, making the chain rule the ideal choice. Using the chain rule, we get $P'(t) = 5(3t^2-1)^4 cdot 6t = 30t(3t^2-1)^4$. While expanding $(3t^2-1)^5$ (option A) would eventually work, it would require expanding a fifth-degree binomial and then differentiating many terms, which is unnecessarily tedious. When you see a function raised to a power, especially a high power, the chain rule is almost always the most efficient approach.

This problem asks for the technique to differentiate a quotient involving a composite trigonometric function. The function $p(x)=rac{ an(2x)}{x^2}$ is a quotient, so the quotient rule is needed. Additionally, $ an(2x)$ requires the chain rule. Using both: $p'(x) = rac{x^2 cdot 2sec^2(2x) - an(2x) cdot 2x}{x^4} = rac{2xsec^2(2x) - 2 an(2x)}{x^3}$. Option D's suggestion to use only the chain rule ignores the quotient structure entirely. When differentiating quotients where one part is composite, combine the quotient rule with the chain rule as needed for each component.

This problem requires selecting the most efficient technique for a logarithm of a quotient. The function $C(x)=lnleft(rac{x^2+1}{x} ight)$ can be simplified using logarithm properties: $lnleft(rac{x^2+1}{x} ight) = ln(x^2+1) - ln(x)$. After this simplification, basic derivative rules give: $C'(x) = rac{2x}{x^2+1} - rac{1}{x} = rac{2x^2 - (x^2+1)}{x(x^2+1)} = rac{x^2-1}{x(x^2+1)}$. Option A would work but requires unnecessary complexity with nested rules. When differentiating logarithms of quotients or products, first use logarithm properties to simplify, then apply basic derivative rules for the most efficient solution.

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 $f'(x) = rac{1}{2}(5x^3-2x+7)^{-1/2} cdot (15x^2-2) = rac{15x^2-2}{2sqrt{5x^3-2x+7}}$. Option E's suggestion to split the square root as a product is mathematically incorrect since $sqrt{a+b}

eq 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.

Selecting the appropriate differentiation technique is a key skill in calculus, ensuring efficiency and accuracy in finding derivatives. For C(x) = (3x² - 5x + $1)^7$, the chain rule combined with the power rule is most efficient because the function is a composition where the outer function is a power and the inner is a polynomial. This method allows us to differentiate the outer function while multiplying by the derivative of the inner function, avoiding complex expansions. It directly yields C'(x) = 7(3x² - 5x + $1)^6$ (6x - 5), which is straightforward and minimizes errors. While expanding the polynomial completely first might seem viable, it would result in a high-degree polynomial that's tedious to differentiate term by term. When choosing a technique, assess the function's structure to prioritize rules like the chain rule for compositions over methods that increase complexity.

Selecting the appropriate differentiation technique is a key skill in calculus, ensuring efficiency and accuracy in finding derivatives. For f(x) = (x² + $1)^{3/2}$ (x - $4)^2$, using the product rule with the chain rule is most appropriate because it's a product of two terms, each requiring the chain rule for their powers. This involves differentiating each factor: the first uses chain on the 3/2 power, and the second on the square, then applying the product rule. It efficiently combines these to find f'(x) without full expansion. Differentiating by expanding completely before would create a messy polynomial with high degrees, increasing error risk. When functions are products of composites, integrate the product rule with chain applications for a balanced and effective approach.

Selecting the appropriate differentiation technique is a key skill in calculus, ensuring efficiency and accuracy in finding derivatives. For $R(x) = \frac{\sin x}{x^2 + 1}$, the quotient rule is most appropriate because the function is a quotient of sin x and a polynomial, allowing direct application of $(\frac{\text{num}' \cdot \text{den} - \text{num} \cdot \text{den}'}{\text{den}^2})$. This efficiently incorporates the derivative of sin x as cos x and the polynomial's derivative. It yields $R'(x) = \frac{\cos x (x^2 + 1) - \sin x (2x)}{(x^2 + 1)^2}$, which is straightforward. Using the power rule only would fail completely since the function involves a trigonometric numerator, not just powers. Match the differentiation technique to the function's primary structure, such as quotient for divisions, to ensure minimal rework.