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 94

Chapter 3, Problem 94

Given that p(x) = (5e^x+10x⁵+20x³+100x²+5x+20) ⋅ (10x⁵+40x³+20x²+4x+10), find p′(0) without computing p′(x).

Verified step by step guidance

Step 1: Recognize that p(x) is a product of two functions, say f(x) = 5e^x + 10x^5 + 20x^3 + 100x^2 + 5x + 20 and g(x) = 10x^5 + 40x^3 + 20x^2 + 4x + 10.

Step 2: Use the product rule for differentiation, which states that if p(x) = f(x)g(x), then p'(x) = f'(x)g(x) + f(x)g'(x).

Step 3: Evaluate f(0) and g(0) by substituting x = 0 into f(x) and g(x).

Step 4: Find f'(0) and g'(0) by differentiating f(x) and g(x) and then substituting x = 0 into the derivatives.

Step 5: Substitute f(0), g(0), f'(0), and g'(0) into the product rule expression for p'(0) to find the value of p'(0).

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.

Product Rule

The Product Rule is a fundamental principle in calculus used to differentiate the product of two functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by p'(x) = u'(x)v(x) + u(x)v'(x). This rule is essential for finding the derivative of p(x) without directly computing p'(x).

Evaluating Derivatives at a Point

To find p′(0), we need to evaluate the derivative of p(x) at x = 0. This involves substituting x = 0 into the derivative expression obtained from the Product Rule. Understanding how to evaluate functions and their derivatives at specific points is crucial for solving problems that require finding instantaneous rates of change.

Exponential and Polynomial Functions

In the given function p(x), we have both exponential (e^x) and polynomial terms (like 10x⁵ and 20x³). Knowing how to differentiate these types of functions is vital, as they behave differently under differentiation. Exponential functions grow rapidly, while polynomial functions have a more predictable growth pattern, which influences the overall behavior of p(x) and its derivative.

Recommended video:

6:13

Exponential Functions

Related Practice

Textbook Question

Composition containing sin x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.

c. g'(π)

Textbook Question

Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.

y = f(x)g(x)

Textbook Question

{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.

a. Find L′(1) and interpret the meaning of this value.

Textbook Question

Derivatives from tangent lines Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.

b. y = f(x) / g(x)

Textbook Question

{Use of Tech} Tangent lines Determine equations of the lines tangent to the graph of y= x√5−x² at the points (1, 2) and (−2,−2). Graph the function and the tangent lines.

Textbook Question

Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.

a. g(1)