Textbook Question
Calculate the derivative of the following functions.
y = sin (4x3 + 3x +1)
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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 35a
Find the derivative function f' for the following functions f.
f(x) =3x²+2x−10; a=1
Verified step by step guidance1
Step 1: Identify the function f(x) = 3x^2 + 2x - 10, which is a polynomial function.
Step 2: Recall the power rule for differentiation, which states that the derivative of x^n is n*x^(n-1).
Step 3: Apply the power rule to each term of the function separately. For the first term, 3x^2, the derivative is 2*3*x^(2-1) = 6x.
Step 4: Apply the power rule to the second term, 2x. The derivative of x is 1, so the derivative of 2x is 2*1 = 2.
Step 5: The derivative of a constant, such as -10, is 0. Combine the derivatives of each term to find f'(x) = 6x + 2.
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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 is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. The derivative is often denoted as f'(x) and represents the slope of the tangent line to the function's graph at any given point.
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Derivatives
Power Rule
The Power Rule is a fundamental technique for finding derivatives of polynomial functions. It states that if f(x) = x^n, where n is a real number, then the derivative f'(x) = n*x^(n-1). This rule simplifies the differentiation process, especially for functions involving powers of x, making it easier to compute derivatives quickly.
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Evaluating Derivatives at a Point
To evaluate the derivative at a specific point, such as a = 1, you first find the derivative function f'(x) and then substitute the value of x with the point of interest. This gives the slope of the tangent line to the function at that specific point, providing insight into the function's behavior at that location.
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Related Practice
Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = x²; a=3
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Textbook Question
A feather dropped on the moon On the moon, a feather will fall to the ground at the same rate as a heavy stone. Suppose a feather is dropped from a height of 40 m above the surface of the moon. Its height (in meters) above the ground after t seconds is s = 40−0.8t². Determine the velocity and acceleration of the feather the moment it strikes the surface of the moon.
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Textbook Question
Derivatives and tangent lines
b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.
f(x) = x²; a=3
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Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
c. It is impossible for the instantaneous velocity at all times a≤t≤b to equal the average velocity over the interval a≤t≤b.
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Textbook Question
Calculate the derivative of the following functions.
y = tan ex
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