Textbook Question
72–76. Tangent lines Find an equation of the line tangent to each of the following curves at the given point.
y = 3x³+ sin x; (0, 0)
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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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.67
Higher-order derivatives Find and simplify y''.
y = 2^x x
Verified step by step guidance1
First, identify the function y = 2^x * x. This is a product of two functions, so we will use the product rule to find the first derivative y'.
Apply the product rule: If y = u * v, then y' = u' * v + u * v'. Here, let u = 2^x and v = x. Find the derivatives u' and v'.
Calculate u': The derivative of 2^x with respect to x is 2^x * ln(2).
Calculate v': The derivative of x with respect to x is 1.
Substitute u', v, u, and v' into the product rule formula to find y'. Then, differentiate y' again to find y'', applying the product rule and chain rule as necessary. Simplify the expression for y''.
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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 taken multiple times. The first derivative, denoted as f'(x), gives the rate of change of the function, while the second derivative, f''(x), provides information about the curvature or concavity of the function. Understanding how to compute these derivatives is essential for analyzing the behavior of functions in calculus.
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Higher Order Derivatives
Product Rule
The product rule is a fundamental differentiation technique used when finding the derivative of a product of two functions. If u(x) and v(x) are two differentiable functions, the product rule states that the derivative of their product is given by u'v + uv'. This rule is crucial for differentiating functions like y = 2^x * x, where both components depend on x.
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Exponential Functions
Exponential functions are functions of the form f(x) = a^x, where a is a constant and x is the variable. These functions have unique properties, such as their derivatives being proportional to the function itself. In the context of the given function y = 2^x * x, recognizing the behavior of the exponential component is vital for accurately computing its derivatives.
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Related Practice
Textbook Question
9–61. Evaluate and simplify y'.
y = x²+2x tan^−1(cot x)
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Textbook Question
Use the given graphs of f and g to find each derivative. <IMAGE>
b. d/dx (f(x)g(x)) |x=1
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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?
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Textbook Question
Evaluate and simplify y'.
y = ln |sec 3x|
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Textbook Question
Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
y =√x³+x−1 at y=3
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