Use formulas (1) and (2) and the power rule to find the derivatives of the following functions. f ( x ) = 3 x + 7 Derivative of Linear Function If f ( x ) = m x + b , then we have f ' ( x ) = m . ( 1 ) Constant Rule The derivative of a constant function f ( x ) = b is zero. That is, f ' ( x ) = 0 . ( 2 ) Power Rule Let r be any number, and let f ( x ) = x r . Then f ' ( x ) = r x r − 1 .
Use formulas (1) and (2) and the power rule to find the derivatives of the following functions. f ( x ) = 3 x + 7 Derivative of Linear Function If f ( x ) = m x + b , then we have f ' ( x ) = m . ( 1 ) Constant Rule The derivative of a constant function f ( x ) = b is zero. That is, f ' ( x ) = 0 . ( 2 ) Power Rule Let r be any number, and let f ( x ) = x r . Then f ' ( x ) = r x r − 1 .
Find an expression for the first two derivatives of f(x)
Find an equation for the function f that has the given derivative and whose graph passes through the given point.
f'(x) = -2x√√√8 - x²
(2,7)
f(x) =
Derivative
Point
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Find the derivative of the function.
2x³ + 2x²
X
f(x)
Step 1
The function f(x) can be simplified to make the algebraic process of finding the derivative easier.
f(x) = 2
f'(x): =
Step 2
Use the rule for the sum of derivatives to differentiate the separate terms of f(x).
d
-(2
dx
Step 3
Use the constant multiple for each term.
d
(2x²) = 2(
dx
dx
d
dx
(2x) = 2.
dx
d
II
dx
= 2(
II
d
d (2x) = 2x (x)
dx
dx
d
+ 2 x
-x
dx
Step 4
Use the power rule to obtain the derivative of the individual terms.
d
(2x²) = 2(x²)
dx
+
X
x
-(2x
dx
2x
Chapter 1 Solutions
Pearson eText for Calculus & Its Applications -- Instant Access (Pearson+)
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY