Textbook Question
5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = sin⁵x
3. Techniques of Differentiation
The Chain Rule
3:35 minutesProblem 45
Textbook Question
Calculate the derivative of the following functions.
y = (2x6 - 3x3 + 3)25
Verified step by step guidance1
Step 1: Recognize that the function y = (2x^6 - 3x^3 + 3)^25 is a composite function, which means we will need to use the chain rule to find its derivative.
Step 2: Identify the outer function and the inner function. Here, the outer function is u^25, where u = 2x^6 - 3x^3 + 3, and the inner function is u = 2x^6 - 3x^3 + 3.
Step 3: Differentiate the outer function with respect to the inner function u. The derivative of u^25 with respect to u is 25u^24.
Step 4: Differentiate the inner function u with respect to x. The derivative of 2x^6 is 12x^5, the derivative of -3x^3 is -9x^2, and the derivative of the constant 3 is 0.
Step 5: Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us the derivative of y with respect to x as dy/dx = 25(2x^6 - 3x^3 + 3)^24 * (12x^5 - 9x^2).
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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 at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and chain rule.
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Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of a composite function. If a function y is defined as a composition of two functions, say y = f(g(x)), the chain rule states that the derivative dy/dx is the product of the derivative of the outer function f with respect to g and the derivative of the inner function g with respect to x. This is essential for differentiating functions raised to a power, as seen in the given problem.
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Power Rule
The power rule is a basic rule for finding the derivative of a function of the form y = x^n, where n is a real number. According to this rule, the derivative is given by dy/dx = n*x^(n-1). This rule simplifies the process of differentiation, especially for polynomial functions, and is crucial for handling terms like 2x^6 and -3x^3 in the provided function.
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