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 45

Chapter 3, Problem 45

Calculate the derivative of the following functions.
y = (2x⁶ - 3x³ + 3)²⁵

Verified step by step guidance

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).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

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.

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.

Recommended video:

Guided course

5:50

Power Rules

Related Practice

Textbook Question

Use the definition of the derivative to determine d/dx (√ax+b), where a and b are constants.

Textbook Question

Calculate the derivative of the following functions.

y = (1 + 2 tan u)^4.5

Textbook Question

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.

f(x) = (√x+1)(√x-1)

views

Textbook Question

Derivative calculations Evaluate the derivative of the following functions at the given point.

f(t) = 1/t+1; a=1

views

Textbook Question

Fish length Assume the length L (in centimeters) of a particular species of fish after t years is modeled by the following graph. <IMAGE>

b. What does the derivative tell you about how this species of fish grows?

views

Textbook Question

Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>