How do you differentiate f(x)= (3x+1)^4(2x-3)^3  using the product rule?

Sep 30, 2016

$12 {\left(3 x + 1\right)}^{3} {\left(2 x - 3\right)}^{3} + 6 {\left(2 x - 3\right)}^{2} {\left(3 x + 1\right)}^{4}$

Explanation:

This problem actually involves two rules: the product rule, and the chain rule. But first, let's just focus on the product rule. The general equation for this is:

$\frac{d}{\mathrm{dx}} \left[f \left(x\right) \cdot g \left(x\right)\right] = \frac{d}{\mathrm{dx}} \left[f \left(x\right)\right] \cdot g \left(x\right) + \frac{d}{\mathrm{dx}} \left[g \left(x\right)\right] \cdot f \left(x\right)$

Now, we just plug in everything:

$\frac{d}{\mathrm{dx}} \left[{\left(3 x + 1\right)}^{4}\right]$${\left(2 x - 3\right)}^{3}$ $+ \frac{d}{\mathrm{dx}} \left[{\left(2 x - 3\right)}^{3}\right]$${\left(3 x - 1\right)}^{4}$

The problem arises at the fact that your two functions are actually compositions of functions. For example, ${\left(3 x + 1\right)}^{4}$ is really the composition of the two functions ${x}^{4}$ and $3 x + 1$. Hence, we'll need to use the chain rule to evaluate these derivatives. Let's take each one in turn:

$\frac{d}{\mathrm{dx}} \left[{\left(3 x + 1\right)}^{4}\right]$

So first, we do the outermost function (in this case ${x}^{4}$)'s derivative, then multiply by the derivative of the inner function ($3 x + 1$). So, this gives us:

=> $4 {\left(3 x + 1\right)}^{3} \cdot \frac{d}{\mathrm{dx}} \left(3 x + 1\right)$
=> $4 {\left(3 x + 1\right)}^{3} \cdot 3$
=> $12 {\left(3 x + 1\right)}^{3}$

Now, for the second one:

$\frac{d}{\mathrm{dx}} \left[{\left(2 x - 3\right)}^{3}\right]$

Again, same process. Derivative of the outside, then multiply by derivative of the inside:

=> $3 {\left(2 x - 3\right)}^{2} \cdot \frac{d}{\mathrm{dx}} \left(2 x - 3\right)$
=> $3 {\left(2 x - 3\right)}^{2} \cdot 2$
=> $6 {\left(2 x - 3\right)}^{2}$

We're done! Now, we just go ahead and plug these back into the product rule equation we got for our final answer:

$12 {\left(3 x + 1\right)}^{3} {\left(2 x - 3\right)}^{3} + 6 {\left(2 x - 3\right)}^{2} {\left(3 x + 1\right)}^{4}$

You could foil out some stuff here, but in my opinion, it's really just more work. Easier to leave it as is.

Hope that helped :)