§2.3 The product and quotient rules

Introduction

The product rule

The quotient rule

Find the derivative of

. Because $f$ is a sum of basic functions, we can now quickly say that $f'(x) = 77x^{10} - 4 \cdot 9^x \ln(9) + \pi \cos(x) + \sqrt{3} \sin(x)$.

Use the product rule to differentiate $P(z) = z^3 \cdot \cos(z)$.

Use the quotient rule to differentiate $Q(t) = \dfrac{\sin(t)}{2^t}$.

If $f$ and $g$ are differentiable, the product rule says that the derivative of $f(x) \cdot g(x)$ equals

Let $f(x) = e^x \sin(x)$. Which expression is $f'(x)$?

For $R(x) = \frac{x^2 - 2x - 8}{x^2 - 9}$, the slope of the tangent line at $x = 0$ is

For $f(x) = e^x \sin(x)$, compute the tangent slope $f'(1)$ rounded to three decimals.

A camera flash has intensity $I(t) = \frac{100t}{e^t}$ candles with $t$ in milliseconds. Compute $I'(2)$ rounded to three decimals.

The quotient rule formula for $\frac{f}{g}$ applies at every point where $g(x) \ne 0$.