We work through several examples illustrating how to use the product rule (also known as “Leibniz‘s rule”) and the quotient rule. Several examples are given at the end to practice with.

## The Product Rule

We being with the product rule for find the derivative of a product of functions. The rule can be generalized to more than the product of two functions.

**Theorem**. (** Product Rule**) If $f= g h$ for any differentiable functions $g$ and $h$, then \begin{equation} f’=g’h+g h’. \end{equation}

**Example**. Find the derivative of the function \begin{equation} f(x)=\left(2x^3+3x\right)\left(x^2-3\right). \end{equation}

**Solution**. We use the product rule with $$ g(x)=2x^3+3x\qquad h(x)=x^2-3 $$ and $f(x)=g(x) h(x).$ We find that \begin{align} f'(x) & =g'(x)h(x)+g(x)h'(x) = \left(2x^3+3x\right)’h(x)+g(x)\left(x^2-3\right)’ \\ & \qquad = \left(6x^2+3\right)\left(x^2-3\right)+\left(2x^3+3x\right)(2x) =10 x^4-9 x^2-9 \end{align} as needed.

**Example**. Find the derivative of the function \begin{equation} f(x) = \left(x^2+3x\right) (3-x) \left(x^4+5x-2\right). \end{equation}

**Solution**. We use the product rule with $$ g(x)=\left(x^2+3x\right)(3-x), \qquad h(x)=x^4+5x-2 $$ and $f(x)=g(x) h(x).$ We find that \begin{align} f'(x) & =g'(x)h(x)+g(x)h'(x) \\ & =\left[\left(x^2+3x\right)(3-x) \right]’ h(x) + \left(x^2+3x\right)(3-x)\left(x^4+5x-2\right)’ \\ & = \left[\left(x^2+3x\right)(3-x)\right]’\left(x^4+5x-2\right)+\left(x^2+3x\right)(3-x) \left(4x^3+5\right) \end{align} Since \begin{align} \left[\left(x^2+3x\right)(3-x)\right]’ & = \left(x^2+3x\right)'(3-x) + \left(x^2+3x\right)(3-x)’ \\ & =(2x+3)(3 -x) + \left(x^2+3x\right)(-1) = 9-3 x^2 \end{align} Thus, $ f'(x) =\left(9-3 x^2 \right)\left(x^4+5x-2\right)+\left(x^2+3x\right)(3-x)\left(4x^3+5\right) $ which can be written as \begin{equation} f'(x)=-7 x^6+45 x^4-20 x^3+6 x^2+90 x-18. \end{equation} as desired.

## The Quotient Rule

The quotient rule is a theorem for finding the derivative of a function which can be written as the ratio of two differentiable functions.

**Theorem**. (** Quotient Rule**) If $f= g /h$ for any differentiable functions $g$ and $h$, then \begin{equation} f’=\frac{h g’-g h’}{h^2}. \end{equation}

**Example**. Find the derivative of the function \begin{equation} f(x)=\frac{\left(x^3-3x\right)\left(x^2-3\right)}{(x-4)x^2}. \end{equation}

**Solution**. We use the quotient rule with $g(x)=\left(x^3-3x\right)\left(x^2-3\right)$ and $h(x)=(x-4)x^2.$ But first we compute $g'(x)=5 x^4-18 x^2+9$ and $h'(x)=3 x^2-8 x.$ Thus, \begin{align} f'(x) & =\frac{\left((x-4)x^2\right)\left(5 x^4-18 x^2+9\right)-\left(x^3-3x\right)\left(x^2-3\right)\left(3 x^2-8 x\right)}{(x-4)^2x^4} \end{align} which can be written as \begin{equation} f'(x)=\frac{9}{4 x^2}-\frac{169}{4 (x-4)^2}+4+2x=\frac{2 \left(18-9 x+12 x^2-6 x^4+x^5\right)}{(-4+x)^2 x^2}. \end{equation}

as needed.

**Example**. Find the first and second derivatives of the function \begin{equation} f(x)=\frac{a x+b}{c x+d}. \end{equation}

**Solution**. Using the product rule with $f(x)=(a x+b)(c x+d)^{-1}$ we find \begin{equation} f'(x)=\frac{a}{d+c x}-\frac{c (b+a x)}{(d+c x)^2}. \end{equation} Using the quotient rule with $$ f(x)=g(x)/h(x), \qquad g(x)=a x+b, $$ and $h(x)=c x+d$. We find that \begin{equation} f'(x)=\frac{a d-b c}{(d+c x)^2}. \end{equation} Notice that the second expression for $f’$ is easier to work with.

## Exercises on Product Rule and Quotient Rule

**Exercise**. Determine $a$ so that $f’\left(\pm \sqrt{\frac{2}{3}}\right)=0$ given $\displaystyle f(x)=x^3-2a x+1.$

**Exercise**. Determine $a$ so that $f’\left(1\pm \sqrt{2}\right)=0$ given $\displaystyle f(x)=\frac{x^2+1}{x-a}.$

**Exercise**. Find the first derivative and the second derivative for each of the following.

$(1) \quad \displaystyle y=\frac{x-1}{x^2+x+1}$

$(2) \quad \displaystyle y=(x-1)\left(x^2+x+1\right)$

$(3) \quad \displaystyle y=\frac{2x+5}{3x-2}$

$(4) \quad \displaystyle f(t)=\frac{t^2-1}{t^2+t-2}$

**Exercise**. Find the derivatives of all orders for each of the following.

$(1) \quad \displaystyle u=\frac{\left(x^2+x\right)\left(x^2-x+1\right)}{x^4}$

$(2) \quad \displaystyle p = \frac{q^2+3}{q^{-3}}$

$(3) \quad \displaystyle w=\frac{3z^2}{z}$

$(4) \quad \displaystyle y=\frac{3x^5+x^2}{x^2}$

**Exercise**. Suppose $u$ and $v$ are differentiable functions of $x$ and that $u(1)=2,$ $u'(1)=0,$ $v(1)=5,$ and $v'(1)=-1.$ Find the values of the following at $x=1.$

$(1) \quad \displaystyle \frac{d}{dx}( u v)$

$(2) \quad \displaystyle \frac{d}{dx}\left(\frac{u}{v}\right)$

$(3) \quad \displaystyle \frac{d}{dx}\left(\frac{v}{u}\right)$

$(4) \quad \displaystyle \frac{d}{dx}(7v-2u)$

**Exercise**. The curves $y=x^2+a x+b$ and $y=c x-x^2$ have a common tangent line at the point $(1,0).$ Find $a,$ $b,$ and $c.$

**Exercise**. If gas in a cylinder is maintained at a constant temperature $T,$ the pressure $P$ is related to the volume $V$ by a formula of the form \begin{equation} P=\frac{n R T}{V-n b}-\frac{a n^2}{V^2}, \end{equation} in which $a,$ $b,$ $n,$ and $R$ are constants. Find $\displaystyle \frac{dP}{dV}.$

**Exercise**. Find the first, second, and third derivatives of the following functions.

$(1) \quad \displaystyle y=\left(x^4+3x^2+17x+82\right)^3$

$(2) \quad \displaystyle y=\frac{x^4+3x^2+17x+82}{\sqrt{x}}$