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 y² + x y = cos x, implicit differentiation is most appropriate because the equation isn't solved explicitly for y, requiring differentiation of both sides with respect to x while treating y as a function of x. This method involves applying the chain rule to terms like y² and the product rule to x y, leading to 2y y' + (y + x y') = -sin x. It allows solving for y' without isolating y, which might be difficult or impossible. Differentiating by solving explicitly for y first could be impractical if the equation doesn't yield a simple explicit form. Use implicit differentiation for equations where variables are intertwined, providing a general strategy for related rates and beyond.

Selecting the appropriate differentiation technique is a key skill in calculus, ensuring efficiency and accuracy in finding derivatives. For G(x) = (x² + 1) sin x, the product rule is most appropriate because it's a product of a polynomial and a trigonometric function, enabling G'(x) = 2x sin x + (x² + 1) cos x. This direct application handles each part's derivative separately and combines them efficiently. It's simpler than alternatives, avoiding unnecessary manipulations. Using the quotient rule would be incorrect since the function isn't a quotient. Identify products early and apply the product rule to maintain simplicity, especially when combining algebraic and transcendental functions.

Selecting the appropriate differentiation technique is a key skill in calculus, ensuring efficiency and accuracy in finding derivatives. For P(t) = $e^{(5t - 1)^2}$, the chain rule is most appropriate because it's an exponential function with a composed exponent, requiring differentiation of the outer exponential and then the inner quadratic. This method efficiently chains the derivatives: the derivative of $e^u$ is $e^u$ times u', where u = (5t - $1)^2$, and u' requires another chain application. It produces P'(t) = $e^{(5t - 1)^2}$ * 2(5t - 1) * 5, which is precise without extra steps. Logarithmic differentiation would add unnecessary complexity by taking the natural log first, which isn't needed for exponentials. For nested functions, apply the chain rule iteratively to handle each layer systematically.

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: $R'(x) = 1 cdot ln(4x+1) + x cdot rac{1}{4x+1} cdot 4 = ln(4x+1) + rac{4x}{4x+1}$. 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.

This problem involves selecting the technique for a product of two different transcendental functions. The function $v(x)=\ln(x^2+1)\cdot\arctan x$ is a product where each factor is a different type of function: a logarithm of a polynomial and an inverse trigonometric function. The product rule is necessary, giving $\ln(x^2+1)\cdot\frac{d}{dx}[\arctan x] + \arctan x\cdot\frac{d}{dx}[\ln(x^2+1)]$. Note that differentiating $\ln(x^2+1)$ requires the chain rule. Using the chain rule only (option B) would be incorrect because this is a product of two separate functions, not a composition. For products of transcendental functions, apply the product rule and use the chain rule within each factor as needed.

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.