How do you use the Product Rule to find the derivative of (7x^4 + 2x^6) sin(7x)?

Aug 9, 2015

${y}^{'} = {x}^{3} \cdot \left[4 \cdot \left(3 {x}^{2} + 7\right) \cdot \sin \left(7 x\right) + 7 x \cdot \left(2 {x}^{2} + 7\right) \cdot \cos \left(7 x\right)\right]$

Explanation:

The product rule allows you to differentiate functions that take the form

$y = f \left(x\right) \cdot g \left(x\right)$

by using the formula

$\textcolor{b l u e}{\frac{d}{\mathrm{dx}} \left(y\right) = \left[\frac{d}{\mathrm{dx}} \left(f \left(x\right)\right)\right] \cdot g \left(x\right) + f \left(x\right) \cdot \frac{d}{\mathrm{dx}} \left(g \left(x\right)\right)}$

In your case, you can think of the function as being

$y = {\underbrace{\left(2 {x}^{6} + 7 {x}^{4}\right)}}_{\textcolor{b l u e}{f \left(x\right)}} \cdot {\underbrace{\sin \left(7 x\right)}}_{\textcolor{g r e e n}{g \left(x\right)}}$

This means that you can write

$\frac{d}{\mathrm{dx}} \left(y\right) = \left[\frac{d}{\mathrm{dx}} \left(2 {x}^{6} + 7 {x}^{4}\right)\right] \cdot \sin \left(7 x\right) + \left(2 {x}^{6} + 7 {x}^{4}\right) \cdot \frac{d}{\mathrm{dx}} \left(\sin \left(7 x\right)\right)$

To differentiate $\sin \left(7 x\right)$, you're going to use the chain rule for $\sin u$, with $u = 7 x$

$\frac{d}{\mathrm{dx}} \left(\sin u\right) = \left[\frac{d}{\mathrm{du}} \cdot \left(\sin u\right)\right] \cdot \frac{d}{\mathrm{dx}} \left(u\right)$

$\frac{d}{\mathrm{dx}} \left(\sin u\right) = \cos u \cdot \frac{d}{\mathrm{dx}} \left(7 x\right)$

$\frac{d}{\mathrm{dx}} \left(\sin \left(7 x\right)\right) = \cos \left(7 x\right) \cdot 7$

This means that your target derivative will be

${y}^{'} = \left(12 {x}^{5} + 28 {x}^{3}\right) \cdot \sin \left(7 x\right) + \left(2 {x}^{6} + 7 {x}^{4}\right) \cdot 7 \cos \left(7 x\right)$

${y}^{'} = 4 {x}^{3} \cdot \left(3 {x}^{2} + 7\right) \cdot \sin \left(7 x\right) + 7 {x}^{4} \cdot \left(2 {x}^{2} + 7\right) \cdot \cos \left(7 x\right)$

${y}^{'} = \textcolor{g r e e n}{{x}^{3} \cdot \left[4 \cdot \left(3 {x}^{2} + 7\right) \cdot \sin \left(7 x\right) + 7 x \cdot \left(2 {x}^{2} + 7\right) \cdot \cos \left(7 x\right)\right]}$