§2.3 The product and quotient rules

Introduction

The product rule

The quotient rule

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

Let $f(t) = 2t^2$ and $g(t) = t^3 + 4t$. Which pair gives the derivatives $f'(t)$ and $g'(t)$?

Which derivative rules suffice to differentiate $f(x) = 7x^{11} - 4 \cdot 9^x + \pi\sin(x) - \sqrt{3}\cos(x)$?

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

Let $m(w) = 3w^{17}4^w$. Which expression is $m'(w)$?

Let $h(t) = (\sin(t) + \cos(t))t^4$. Which expression is $h'(t)$?

Let $g(x) = (x^2 + x)2^x$. Since $g(-1) = 0$ and $g'(-1) = -\frac{1}{2}$, the tangent line approximation at $(-1, g(-1))$ is

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

Let $r(z) = \frac{3^z}{z^4 + 1}$. Which expression is $r'(z)$?

Let $v(t) = \frac{\sin(t)}{\cos(t) + t^2}$. Which expression is $v'(t)$?

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 same flash satisfies $I'(0.5) \approx 30.3$, $I'(2) \approx -13.5$, and $I'(5) \approx -2.7$ candles per millisecond. What story do these numbers tell?

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