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 21

Chapter 3, Problem 21

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 = e^4x²+1

Verified step by step guidance

Step 1: Identify the composite function structure. The given function is $ y = e^{4x^2 + 1} $. This can be seen as a composition of two functions.

Step 2: Define the inner function $ u = g(x) $. Here, choose $ u = 4x^2 + 1 $. This simplifies the expression inside the exponent.

Step 3: Define the outer function $ y = f(u) $. With $ u = 4x^2 + 1 $, the outer function becomes $ y = e^u $.

Step 4: Differentiate the outer function with respect to $ u $. The derivative of $ y = e^u $ with respect to $ u $ is $ \frac{dy}{du} = e^u $.

Step 5: Differentiate the inner function with respect to $ x $. The derivative of $ u = 4x^2 + 1 $ with respect to $ x $ is $ \frac{du}{dx} = 8x $. Now, use the chain rule to find $ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = e^u \cdot 8x $. Substitute back $ u = 4x^2 + 1 $ to get $ \frac{dy}{dx} = e^{4x^2 + 1} \cdot 8x $.

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Key Concepts

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

Composite Functions

A composite function is formed when one function is applied to the result of another function. In the context of the question, we need to identify an inner function g(x) and an outer function f(u) such that the overall function can be expressed as y = f(g(x)). Understanding how to decompose a function into its inner and outer components is crucial for differentiation.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if y = f(g(x)), then the derivative dy/dx can be calculated as dy/dx = f'(g(x)) * g'(x). This rule allows us to find the derivative of complex functions by breaking them down into simpler parts, making it essential for solving the given problem.

Exponential Functions

Exponential functions are mathematical functions of the form y = a * e^(bx), where e is the base of natural logarithms. In the given function y = e^(4x² + 1), recognizing the structure of the exponential function is important for identifying the outer function. Understanding the properties of exponential functions, including their derivatives, is key to applying the chain rule effectively.

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Exponential Functions

Related Practice

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 = √7x-1

Textbook Question

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t² + 32t + 48.

Determine the velocity v of the stone after t seconds.

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 = e^√x

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 = tan 5x²

Textbook Question

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

On what intervals is the speed increasing?

f(t) = 6t³ + 36t² - 54t; 0 ≤ t ≤ 4

Textbook Question

Position, velocity, and acceleration Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

a. Graph the position function.

$f (t) = 6 t^{3} + 36 t^{2} - 54 t; 0 \leq t \leq 4$

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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 = e^4x²+1

Verified video answer for a similar problem:

Key Concepts

Composite Functions

Chain Rule

Exponential Functions

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 = e^4x²+1