3.2: The Derivative as a Function

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Dec 19, 2023
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- 3.1: Defining the Derivative
- 3.3: Differentiation Rules

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Use the definition of the derivative to differentiate $f(x) = 2$ .
Use the definition of the derivative to differentiate $g(x) = 3x - 4$ .
Use the definition of the derivative to differentiate $h(x) = x^2$ .
Use the definition of the derivative to differentiate $y = 2x^2 - x + 4$ .
Use the definition of the derivative to differentiate $f(t) = t^3 - 5$ .
Use the definition of the derivative to differentiate $g(t) = \sqrt{t}$ .
Use the definition of the derivative to differentiate $h(t) = \sqrt{2 - 3t}$ .
Use the definition of the derivative to differentiate $y = \dfrac{1}{x - 2}$ .
Use the definition of the derivative to differentiate $f(x) = \dfrac{1}{x^2}$ .
Use the definition of the derivative to differentiate $g(x) = \dfrac{1}{\sqrt{x + 1}}$ .
Use the definition of the derivative to differentiate $h(x) = \left\{\begin{align*} -x, && x \leq 0, \\ x^2, && x > 0. \end{align*}\right.$
[Hint: Be careful at $x = 0$ .]
Use the definition of the derivative to differentiate $f(x) = \left\{\begin{align*} -x, && x \leq 0, \\ x^2 - x, && x > 0. \end{align*}\right.$
[Hint: Be careful at $x = 0$ .]
The graph of $g(x)$ is shown below. Sketch the graph of $g'(x)$ .
$Graph of g(x) = 2.$
The graph of $h(x)$ is shown below. Sketch the graph of $h'(x)$ .
$Graph of h(x) = -0.5x + 1.$
The graph of $f(x)$ is shown below. Sketch the graph of $\dfrac{df}{dx}$ .
$Graph of f(x) = 0.5x^2 - x - 3.$
The graph of $g(x)$ is shown below. Sketch the graph of $\dfrac{dg}{dx}$ .
$Graph of g(x) = -x * (x-1) * (x-3).$
The graph of $h(x)$ is shown below. Sketch the graph of $\dfrac{dh}{dx}$ .
$Graph of h(x) = 0.2 * (x-2)^2 * (x+2)^2$
The graph of $f(x)$ is shown below. Sketch the graph of $f'(x)$ .
$Graph of f(x) = -0.6 * (0.25x^4 - 1.5x^2 - 2x).$
The graph of $g(x)$ is shown below. Sketch the graph of $g'(x)$ .
$Graph of g(x) = 3/(x^2 + 1).$
The graph of $h(x)$ is shown below. Sketch the graph of $h'(x)$ .
$Graph of h(x) = sin(x) on the interval from -2*pi to 2*pi.$
The graph of $f(x)$ is shown below. Sketch the graph of $\dfrac{df}{dx}$ .
$Graph of f(x) = |x|.$
The graph of $g(x)$ is shown below. Sketch the graph of $\dfrac{dg}{dx}$ .
$Graph of g(x) = 2.5*x^(1/3).$
The graph of $h(x)$ is shown below. Sketch the graph of $\dfrac{dh}{dx}$ .
$Graph of the function h(x). The function h(x) = x-1 for x < 0 and h(x) = x+1 for x >= 0.$
Use the graph of f(x) below to answer the following.
1. List all values of $x$ for which $f(x)$ is not defined.
2. List all values of $x$ for which $\displaystyle \lim_{t \rightarrow x} f(t)$ does not exist.
3. List all values of $x$ for which $f(x)$ is not continuous.
4. List all values of $x$ for which $f(x)$ is not differentiable.
$The graph of the piecewise function f(x). Function f(x) = x^2 + 6.1097x + 9.7677 for x < -3; function f(x) = 1/|x|^(3/4) for -3 < x < 0; function f(x) = 1 at x = 0; function f(x) = 1/|x|^(3/4) for 0 < x <= 2; function f(x) = -4(x-2.25)^2 + 0.8446 for 2 < x < 3; and f(x) = -2(x-4)^(1/3) for x >= 3.$

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