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 How to Use the Definition of the Derivative. Visual Explanation with color coded examples - 22 Practice Problems explained step by step with interactive problems, showing all work.

How to Use the Definition of the Derivative:
Practice Problems

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Problem 1

Suppose f(x)=x2+3x. Evaluate f(1) using the version of the derivative definition shown below.

f(x)=lim

Problem 2

Find \displaystyle \frac d {dx}\left(-x^3\right) using the version of the derivative definition shown below.

\frac{df}{dx} = \displaystyle\lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}

Problem 3

Find \displaystyle \frac d {dx} \left(\sqrt{x+3}\right) using the version of the derivative definition shown below.

\frac{df}{dx} = \displaystyle\lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}

Problem 4

Suppose f(x) = \sqrt{3x}. Find f'(12) using the version of the derivative definition shown below.

f'(x) = \displaystyle\lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}

Problem 5

Suppose f(t) = \frac 1 {t+4}. Find f'(2) using the version of the derivative definition shown below.

f'(t) = \displaystyle\lim_{\Delta t \to 0} \frac{f(t+\Delta t) - f(t)}{\Delta t}

Problem 6

Find \displaystyle \frac d {dx}\left(\frac 2 {5x}\right) using the version of the derivative definition shown below.

\frac{df}{dx} = \displaystyle\lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}

Problem 7

Find \displaystyle \frac d {d\theta} \left(\cos \theta\right) when \theta is in radians. Use the version of the derivative definition shown below.

\frac{df}{d\theta} = \displaystyle\lim_{\Delta \theta \to 0} \frac{f(\theta+\Delta \theta) - f(\theta)}{\Delta \theta}

Problem 8

Suppose f(x) = e^{2x}. Find f'(0) using the version of the derivative definition shown below.

f'(x) = \displaystyle\lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}

Problem 9

Find \displaystyle \frac d {dx} \left(4x + 7\right) using the version of the definition of the derivative shown below.

\frac{df}{dx} = \displaystyle\lim_{h\to 0} \frac{f(x + h) - f(x)} h

Problem 10

Find \displaystyle \frac d {dx} \left(x^2 + 6\right) using the version of the definition of the derivative shown below.

\frac{df}{dx} = \displaystyle\lim_{h\to 0} \frac{f(x + h) - f(x)} h

Problem 11

Suppose f(x) = \sqrt{2x+1}. Find f'(0) using the version of the definition of the derivative shown below.

f'(x) = \displaystyle\lim_{h\to 0} \frac{f(x + h) - f(x)} h

Problem 12

Find \displaystyle \frac d {dx}\left(\sqrt{1-5x}\right) using the version of the definition of the derivative shown below.

\frac{df}{dx} = \displaystyle\lim_{h\to 0} \frac{f(x + h) - f(x)} h

Problem 13

Find \displaystyle \frac d {dx}\left( \frac 4 {2x+3}\right) using the version of the definition of the derivative shown below.

\frac{df}{dx} = \displaystyle\lim_{h\to 0} \frac{f(x + h) - f(x)} h

Problem 14

Suppose \displaystyle f(x) = \frac 1 x - \frac 1 {x+2}. Find f'(3) using the version of the definition of the derivative shown below.

f'(x) = \displaystyle\lim_{h\to 0} \frac{f(x + h) - f(x)} h

Problem 15

Suppose f(x) = \sin(\pi x) with x in radians. Find f'(4) using the version of the definition of the derivative shown below.

f'(x) = \displaystyle\lim_{h\to 0} \frac{f(x + h) - f(x)} h

Problem 16

Suppose f(x) = \cos(6x) with x in radians. Find f'\left(\frac\pi 6\right) using the version of the definition of the derivative shown below.

f'(x) = \displaystyle\lim_{h\to 0} \frac{f(x + h) - f(x)} h

Problem 17

Suppose f(x) = x^2 + 2x + 3. Evaluate f'\left(\frac 1 2\right) using the version of the definition of the derivative shown below.

f'(a) = \displaystyle\lim_{x\to a} \frac{f(x) - f(a)}{x-a}

Problem 18

Suppose f(x) = 4x^3 -2. Evaluate f'(-4) using the version of the definition of the derivative shown below.

f'(a) = \displaystyle\lim_{x\to a} \frac{f(x) - f(a)}{x-a}

Problem 19

Suppose f(x) = \sqrt{9x-2}. Find f'(1) using the version of the definition of the derivative shown below.

f'(a) = \displaystyle\lim_{x\to a} \frac{f(x) - f(a)}{x-a}

Problem 20

Suppose f(x) = \frac 1 {\sqrt{5x}}. Find f'(45) using the version of the definition of the derivative shown below.

f'(a) = \displaystyle\lim_{x\to a} \frac{f(x) - f(a)}{x-a}

Problem 21

Suppose \displaystyle f(x) = \frac 1 {6x + 5}. Find f'(2) using the version of the definition of the derivative shown below.

f'(a) = \displaystyle\lim_{x\to a} \frac{f(x) - f(a)}{x-a}

Problem 22

Suppose \displaystyle f(x) = \frac 3 {x^2+1}. Find f'(3) using the version of the definition of the derivative shown below.

f'(a) = \displaystyle\lim_{x\to a} \frac{f(x) - f(a)}{x-a}

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