Textbook Question
Find d/dx(ln√x²+1).
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.5.25
Find the derivative of the following functions.
y = e^-x sin x
Verified step by step guidance1
Step 1: Identify the function y = e^(-x) * sin(x) as a product of two functions, u(x) = e^(-x) and v(x) = sin(x). This suggests using the product rule for differentiation.
Step 2: Recall the product rule for differentiation, which states that if y = u(x) * v(x), then the derivative y' = u'(x) * v(x) + u(x) * v'(x).
Step 3: Differentiate u(x) = e^(-x). The derivative of e^(-x) with respect to x is -e^(-x).
Step 4: Differentiate v(x) = sin(x). The derivative of sin(x) with respect to x is cos(x).
Step 5: Apply the product rule: y' = (-e^(-x)) * sin(x) + e^(-x) * cos(x). Combine these results to express the derivative of the original function.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas 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 represents the slope of the tangent line to the curve of the function at any given point. In calculus, derivatives are fundamental for understanding rates of change and are used extensively in optimization problems.
Recommended video:
05:44
Derivatives
Product Rule
The product rule is a formula used to find the derivative of 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 u'v + uv'. This rule is essential when differentiating functions that are products of simpler functions, such as the given function y = e^(-x) sin(x).
Recommended video:
05:18The Product Rule
Exponential and Trigonometric Functions
Exponential functions, like e^(-x), and trigonometric functions, such as sin(x), have specific derivatives that are crucial for differentiation. The derivative of e^x is e^x, and the derivative of sin(x) is cos(x). Understanding these derivatives is vital for applying the product rule effectively in the context of the given function.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
√x⁴+y² = 5x+2y³
Textbook Question
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = x^In x
Textbook Question
27–76. Calculate the derivative of the following functions.
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).
f(x) = In(3x + 1)⁴
Textbook Question
A 12-ft ladder is leaning against a vertical wall when Jack begins pulling the foot of the ladder away from the wall at a rate of 0.2 ft/s. What is the configuration of the ladder at the instant when the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder?