Textbook Question
27–76. Calculate the derivative of the following functions.
3. Techniques of Differentiation
The Chain Rule
7:03 minutesProblem 64
Textbook Question
Calculate the derivative of the following functions.
y = (e^x / x+1)⁸
Verified step by step guidance1
Step 1: Recognize that the function y = \left(\frac{e^x}{x+1}\right)^8 is a composite function, which means we will need to use the chain rule to find its derivative.
Step 2: Let u = \frac{e^x}{x+1}. Then, y = u^8. The chain rule states that \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}. First, find \frac{dy}{du} = 8u^7.
Step 3: Now, find \frac{du}{dx}. Since u = \frac{e^x}{x+1}, use the quotient rule: \frac{du}{dx} = \frac{(x+1)\cdot e^x - e^x \cdot 1}{(x+1)^2}.
Step 4: Simplify \frac{du}{dx} to get \frac{e^x(x+1) - e^x}{(x+1)^2} = \frac{e^x(x+1-1)}{(x+1)^2} = \frac{e^x x}{(x+1)^2}.
Step 5: Substitute \frac{du}{dx} and \frac{dy}{du} back into the chain rule formula: \frac{dy}{dx} = 8u^7 \cdot \frac{e^x x}{(x+1)^2}. Finally, replace u with \frac{e^x}{x+1} to express the derivative in terms of x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve of the function at any given point. The derivative is often denoted as f'(x) or dy/dx and can be calculated using various rules, such as the power rule, product rule, and quotient rule.
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Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when dealing with functions raised to a power, as in the given problem.
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Quotient Rule
The quotient rule is used to find the derivative of a function that is the ratio of two other functions. If y = u/v, where u and v are functions of x, the derivative is given by (v * du/dx - u * dv/dx) / v². This rule is essential for differentiating functions like y = (e^x)/(x+1), where the numerator and denominator are both functions that need to be differentiated separately.
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