Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 62

Chapter 3, Problem 62

Calculate the derivative of the following functions.
y = (f(g(x^m)))^n, where f and g are differentiable for all real numbers and m and n are constants

Verified step by step guidance

Step 1: Recognize that the function y = (f(g(x^m)))^n is a composition of functions and requires the use of the chain rule for differentiation. The chain rule states that if you have a composite function y = h(u(x)), then the derivative y' = h'(u(x)) * u'(x).

Step 2: Apply the chain rule to the outermost function, which is (f(g(x^m)))^n. Let u = f(g(x^m)), then y = u^n. The derivative of y with respect to u is dy/du = n * u^(n-1).

Step 3: Differentiate the inner function u = f(g(x^m)) with respect to x. First, find the derivative of f(g(x^m)) with respect to g(x^m), which is f'(g(x^m)). Then, multiply by the derivative of g(x^m) with respect to x.

Step 4: Differentiate g(x^m) with respect to x. Use the chain rule again, where v = x^m, so g(v) = g(x^m). The derivative of g(v) with respect to v is g'(v), and the derivative of v = x^m with respect to x is m * x^(m-1).

Step 5: Combine all the derivatives using the chain rule. The derivative of y with respect to x is dy/dx = n * (f(g(x^m)))^(n-1) * f'(g(x^m)) * g'(x^m) * m * x^(m-1). This expression represents the derivative of the original function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chain Rule

The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, which in turn is a function of x, then the derivative of y with respect to x is the product of the derivative of y with respect to u and the derivative of u with respect to x. This rule is essential for calculating derivatives of functions like y = (f(g(x^m)))^n.

Product Rule

The Product Rule is a technique for differentiating products of two or more functions. It states that if you have two functions u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is particularly useful when dealing with functions raised to a power, as seen in the expression (f(g(x^m)))^n, where the outer function is raised to the power n.

Power Rule

The Power Rule is a basic differentiation rule that applies to functions of the form y = x^k, where k is a constant. It states that the derivative of y with respect to x is kx^(k-1). This rule is crucial when differentiating terms like (f(g(x^m)))^n, as it allows for the straightforward differentiation of the outer function raised to a power.

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{Use of Tech} Equations of tangent lines

Find an equation of the line tangent to the given curve at a.

y = e^x; a = ln 3

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Textbook Question

Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = 2x² / (3x - 1); a = 1