Product Rule

The product rule is useful for differentiating the product of functions.

It states that for any functions \(u\) and \(v\),

\[ \frac{ d}{dx} uv = u \frac{d}{dx} v + v \frac{d}{dx} u. \]

Or, in Lagrange notation,

\[(uv)' = u'v+v'u.\]

We begin by recalling the definition of differentiation:

\[\frac{d}{dx}f(x) = \lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}.\]

Thus, we have

\[\begin{align} \dfrac{d}{dx}(uv)&=\displaystyle\lim_{h\to 0}\dfrac{u(x+h)v(x+h)-u(x)v(x)}{h}\\ &=\displaystyle\lim_{h\to 0}\dfrac{u(x+h)v(x+h)-u(x+h)v(x)+u(x+h)v(x)-u(x)v(x)}{h}\\ &=\displaystyle\lim_{h\to 0}\left[u(x+h)\dfrac{v(x+h)-v(x)}{h}+v(x)\dfrac{u(x+h)-u(x)}{h}\right]\\ &=\displaystyle\lim_{h\to 0}u(x+h)\cdot\displaystyle\lim_{h\to 0}\dfrac{v(x+h)-v(x)}{h}+v(x)\cdot\displaystyle\lim_{h\to 0}\dfrac{u(x+h)-u(x)}{h}\\ &=u\dfrac{dv}{dx}+v\dfrac{du}{dx}.\ _\square \end{align}\]

The product rule can also be applied to the product of numerous variables. For example,

\[ \frac{ d}{dx} uvw = uv \frac{d}{dx} w + uw \frac{d}{dx} v + vw \frac{d}{dx} u,\]

or in Lagrange notation

\[(uvw)' = uvw' +uwv' + vwu'.\]

What is \( \frac{d}{dx} x e^x?\)

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Applying the product rule with \( u = x \) and \( v = e^x \), we get that

\[ \frac{d}{dx} x e^x = x \frac{d}{dx} e^x + e^x \frac{d}{dx} x = x \times e^x + e^x \times 1 = (x+1) e^x . \]

Hence, \( \frac{d}{dx} x e^x = (x+1) e^x \). \( _\square \)

Find the derivative of

\[f(x) = x^2\sin x.\]

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Since \((x^2)' = 2x \) and \((\sin x)' = \cos x,\)

\[f'(x) = (2x)(\sin x)+(\cos x)(x^2) = 2x\sin x + x^2\cos x.\ _\square\]

What is \( \frac{ d}{dx} c f(x)?\)

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We may recognize this as a basic property of integerals. Applying the product rule with \( u = c\) and \( v = f(x) \), we get that

\[ \frac{d}{dx} c f(x) = c \frac{d}{dx} f(x) + f(x) \frac{d}{dx} c = c f'(x) + f(x) \times 0 = c f'(x). \]

Hence, \( \frac{d}{dx} c f(x) = c f'(x) \). \( _\square \)

Some functions may require the combined use of differentiation rules, such as this one here:

If you are having some trouble, you may want to review the chain rule.