Quick Overview
- In the previous lesson, we established various rules for working with derivatives.
- Rather than use the definition of the derivative each time, we can instead use these "short-cut" rules.
Examples
Example 1
Find $$\displaystyle \frac d {dx}\left(8x -3\right)$$
Step 1Use the rule for linear functions.
We know that the derivative of any linear function is just the slope of the line. Consequently,
$$ \frac d {dx}\left(8x -3\right) = 8 $$
Example 2
Suppose $$f(x) = \sin 3x - 4\cos 7x$$. Find $$f'\left(\frac \pi 2\right)$$
Step 1Find the $$f'(x)$$ using the rules (not the definition).
$$ \begin{align*} f'(x) & = \frac d {dx} \left(\sin 3x\right) - \frac d {dx}\left( 4\cos 7x\right) && \mbox{(Difference Rule)}\\[6pt] & = \frac d {dx} \left(\sin 3x\right) - 4\cdot\frac d {dx}\left( \cos 7x\right) && \mbox{(Coefficient Rule)}\\[6pt] & = 3\cos 3x - 4\cdot\frac d {dx}\left( \cos 7x\right) && \mbox{(Derivative of the Sine)}\\[6pt] & = 3\cos 3x - 4\cdot(-7\sin 7x) && \mbox{(Derivative of the Cosine)}\\[6pt] & = 3\cos 3x +28\sin 7x \end{align*} $$
Step 2Evaluate the derivative at $$x = \pi / 2$$.
$$ \begin{align*} f'\left(\frac \pi 2\right) & = 3\cos\left(3\cdot \frac \pi 2\right) +28\sin\left(7\cdot \frac \pi 2\right)\\[6pt] & = 3\cos\left(\frac{3\pi} 2\right) +28\sin\left(\frac{7\pi} 2\right)\\[6pt] & = 3(0) + 28(-1)\\[6pt] & = -28 \end{align*} $$
Answer$$\displaystyle f'\left(\frac \pi 2\right) = -28$$ when $$f(x) = \sin 3x - 4\cos 7x$$.
