Skip to main content
Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.5.59
Chapter 3, Problem 3.5.59

Find y'' for the following functions.
y = ex sin x

Verified step by step guidance
1
First, identify the function y = e^x * sin(x). This is a product of two functions, so we will use the product rule for differentiation.
Apply the product rule: If y = u * v, then y' = u' * v + u * v'. Here, let u = e^x and v = sin(x).
Differentiate u = e^x to get u' = e^x. Differentiate v = sin(x) to get v' = cos(x).
Substitute these derivatives into the product rule formula: y' = e^x * sin(x) + e^x * cos(x).
Differentiate y' again to find y''. Use the product rule on each term separately: For the first term, differentiate e^x * sin(x) again, and for the second term, differentiate e^x * cos(x). Combine these results to find y''.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

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

Higher-Order Derivatives

Higher-order derivatives refer to the derivatives of a function beyond the first derivative. The second derivative, denoted as y'', measures the rate of change of the first derivative, providing insights into the function's concavity and acceleration. Understanding how to compute higher-order derivatives is essential for analyzing the behavior of functions in calculus.
Recommended video:
02:42
Higher Order Derivatives

Product Rule

The product rule is a fundamental differentiation technique used when finding the derivative of the product of two functions. It states that if u(x) and v(x) are functions, then the derivative of their product is given by u'v + uv'. This rule is particularly important when differentiating functions like y = e^x sin x, which is a product of two distinct functions.
Recommended video:
05:18
The Product Rule

Chain Rule

The chain rule is a method for differentiating composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x is the derivative of y with respect to u multiplied by the derivative of u with respect to x. This rule is crucial when dealing with functions that involve exponentials and trigonometric functions, as seen in y = e^x sin x.
Recommended video:
05:02
Intro to the Chain Rule
Related Practice
Textbook Question

Derivatives Find and simplify the derivative of the following functions.

f(x) = x /x+1

290
views
Textbook Question

15–48. Derivatives Find the derivative of the following functions.

y = 10^x(In 10^x-1)

207
views
Textbook Question

47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

f(x)=tan x; (1,π/4)

148
views
Textbook Question

A projectile is fired vertically upward into the air; its position (in feet) above the ground after t seconds is given by the function s (t). For the following functions, use limits to determine the instantaneous velocity of the projectile at t = a seconds for the given value of a.

s(t) = -16t2 + 128t + 192; a = 2

274
views
Textbook Question

Derivatives Find and simplify the derivative of the following functions.

f(t) = t⁵/³e^t

356
views
Textbook Question

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).


y = 4 log₃(x²−1)

205
views