Textbook Question
What is the derivative of y = e^kx?
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 12
Use the table to find the following derivatives.
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d/dx (f(x) + g(x)) ∣x=1
Verified step by step guidance1
Step 1: Recall the sum rule for derivatives, which states that the derivative of a sum of functions is the sum of their derivatives. In mathematical terms, this is expressed as \( \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x) \).
Step 2: Identify the point at which you need to evaluate the derivative, which is \( x = 1 \) in this problem.
Step 3: Use the table provided in the problem to find the values of \( f'(1) \) and \( g'(1) \). These are the derivatives of \( f(x) \) and \( g(x) \) evaluated at \( x = 1 \).
Step 4: Substitute the values of \( f'(1) \) and \( g'(1) \) from the table into the expression \( f'(1) + g'(1) \).
Step 5: Simplify the expression to find the value of the derivative \( \frac{d}{dx} (f(x) + g(x)) \) at \( x = 1 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
A derivative represents the rate at which a function changes at a given point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or df/dx, and it provides crucial information about the function's behavior, such as its slope and concavity.
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Derivatives
Sum Rule of Derivatives
The Sum Rule states that the derivative of the sum of two functions is equal to the sum of their derivatives. Mathematically, if f(x) and g(x) are differentiable functions, then d/dx (f(x) + g(x)) = f'(x) + g'(x). This rule simplifies the process of finding derivatives when dealing with the addition of functions.
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Evaluating Derivatives at a Point
Evaluating a derivative at a specific point involves substituting the value of that point into the derivative function. For example, to find d/dx (f(x) + g(x)) at x=1, you first compute f'(1) and g'(1) using the derivatives obtained from the Sum Rule, and then add these values together to get the final result.
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Related Practice
Textbook Question
Shrinking square The sides of a square decrease in length at a rate of 1 m/s.
a. At what rate is the area of the square changing when the sides are 5 m long?
Textbook Question
If f′(−2) = 7, find an equation of the line tangent to the graph of f at the point (−2,4).
Textbook Question
The sides of a square decrease in length at a rate of 1 m/s.
b. At what rate are the lengths of the diagonals of the square changing?
Textbook Question
Let F(x) = f(x) + g(x),G(x) = f(x) - g(x), and H(x) = 3f(x) + 2g(x), where the graphs of f and g are shown in the figure. Find each of the following.
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H'(2)
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Textbook Question
Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
h(z) = (z3 + 4z2 + z)(z - 1)