Textbook Question
9–61. Evaluate and simplify y'.
y = e^sin x+2x+1
1
views
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.R.46
9–61. Evaluate and simplify y'.
y = e^6x sin x
Verified step by step guidance1
Step 1: Identify the function y = e^{6x} \(\sin\) x as a product of two functions, u(x) = e^{6x} and v(x) = \(\sin\) x.
Step 2: Apply the product rule for differentiation, which states that if y = u(x)v(x), then y' = u'(x)v(x) + u(x)v'(x).
Step 3: Differentiate u(x) = e^{6x} with respect to x. Use the chain rule: u'(x) = \(\frac{d}{dx}\)[e^{6x}] = 6e^{6x}.
Step 4: Differentiate v(x) = \(\sin\) x with respect to x. The derivative is v'(x) = \(\cos\) x.
Step 5: Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula: y' = 6e^{6x} \(\sin\) x + e^{6x} \(\cos\) x.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differentiation
Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this case, we need to differentiate the function y = e^(6x) sin(x) to find y'. This involves applying rules such as the product rule and the chain rule, which are essential for handling functions that are products of other functions.
Recommended video:
05:53
Finding Differentials
Product Rule
The product rule is a formula used to differentiate products 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'. In the context of the given function y = e^(6x) sin(x), we will apply the product rule to differentiate the exponential function and the sine function together.
Recommended video:
05:18The Product Rule
Chain Rule
The chain rule is a fundamental technique in calculus 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. In this problem, the chain rule will be necessary when differentiating the exponential part e^(6x), as it involves an inner function (6x) that also needs to be differentiated.
Recommended video:
05:02Intro to the Chain Rule
Related Practice
Textbook Question
Evaluate and simplify y'.
y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷
Textbook Question
9–61. Evaluate and simplify y'.
y = e^sin (cosx)
Textbook Question
A jet flying at 450 mi/hr and traveling in a straight line at a constant elevation of 500 ft passes directly over a spectator at an air show. How quickly is the angle of elevation (between the ground and the line from the spectator to the jet) changing 2 seconds later?
Textbook Question
9–61. Evaluate and simplify y'.
y = tan (sin θ)
Textbook Question
9–61. Evaluate and simplify y'.
y = csc⁵ 3x