Derivatives and tangent lines: practice problems

Average rate of change and the derivative

The average rate of change from $x = a$ to $x = b$ is $\frac{f(b) - f(a)}{b - a}$, the slope of the chord.

Example: for $f(x) = x^2$, $\frac{f(1 + h) - f(1)}{h} = 2 + h \to 2$, so $f'(1) = 2$.

Limits such as $\lim_{h \to 0}\frac{f(a + 2h) - f(a - h)}{h}$ are handled by subtracting and adding $f(a)$, giving $3f'(a)$.

Differentiating polynomials

Differentiate term by term; expand products such as $(2x + 1)(x - 3)$ first. Example: $(x^3 - 4x^2 + 5)' = 3x^2 - 8x$.

The same rules apply to other variables: the area of a circle $S = \pi r^2$ has $\frac{dS}{dr} = 2\pi r$, the circumference.

Tangent and normal lines

For a tangent drawn from a point not on the curve, let the point of contact be at $x = t$, write the tangent, and require it to pass through the given point.