3.2: The Derivative as a Function
- Page ID
- 143970
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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 \(g'(x)\).
- The graph of \(h(x)\) is shown below. Sketch the graph of \(h'(x)\).
- The graph of \(f(x)\) is shown below. Sketch the graph of \(\dfrac{df}{dx}\).
- The graph of \(g(x)\) is shown below. Sketch the graph of \(\dfrac{dg}{dx}\).
- The graph of \(h(x)\) is shown below. Sketch the graph of \(\dfrac{dh}{dx}\).
- The graph of \(f(x)\) is shown below. Sketch the graph of \(f'(x)\).
- The graph of \(g(x)\) is shown below. Sketch the graph of \(g'(x)\).
- The graph of \(h(x)\) is shown below. Sketch the graph of \(h'(x)\).
- The graph of \(f(x)\) is shown below. Sketch the graph of \(\dfrac{df}{dx}\).
- The graph of \(g(x)\) is shown below. Sketch the graph of \(\dfrac{dg}{dx}\).
- The graph of \(h(x)\) is shown below. Sketch the graph of \(\dfrac{dh}{dx}\).
- Use the graph of \(f(x)\) below to answer the following.
- List all values of \(x\) for which \(f(x)\) is not defined.
- List all values of \(x\) for which \(\displaystyle \lim_{t \rightarrow x} f(t)\) does not exist.
- List all values of \(x\) for which \(f(x)\) is not continuous.
- List all values of \(x\) for which \(f(x)\) is not differentiable.
- List all values of \(x\) for which \(f(x)\) is not defined.