Product Rule and Quotient Rule

Maybe you are wondering what the product rule is, or are you are trying to get a handle on using the quotient rule? I discuss several examples that will help you understand these theorems. I work through them slowly and show you how to simplify. After you master these rules, you will be ready for more advanced rules, such as the chain rule.

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.

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

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

David A. Smith at Dave4Math

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.

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