# How do you differentiate y = (x^3 + 2)^2(x^5 + 4)^4?

Mar 13, 2018

To differentiate the expression, you could you explicit differentiation or implicit differentiation. I'm going to use implicit differentiation.

#### Explanation:

Using the chain rule in conjunction with the product rule, we get:

$\frac{\mathrm{dy}}{\mathrm{dx}} \left[{\left({x}^{3} + 2\right)}^{2} {\left({x}^{5} + 4\right)}^{4}\right]$

=d/dx[(x^3+2)^2]⋅(x^5+4)^4+(x^3+2)2⋅d/dx[(x^5+4)^4]

=2(x^3+2)⋅d/dx[x^3+2]⋅(x^5+4)^4+4(x^5+4)^3⋅d/dx[x^5+4]⋅(x^3+2)^2

$= 2 \left(\frac{d}{\mathrm{dx}} \left[{x}^{3}\right] + \frac{d}{\mathrm{dx}} \left[2\right]\right) \left({x}^{3} + 2\right) {\left({x}^{5} + 4\right)}^{4} + 4 \left(\frac{d}{\mathrm{dx}} \left[{x}^{5}\right] + \frac{d}{\mathrm{dx}} \left[4\right]\right) {\left(x 3 + 2\right)}^{2} {\left({x}^{5} + 4\right)}^{3}$

$= 2 \left(3 {x}^{2} + 0\right) \left({x}^{3} + 2\right) {\left({x}^{5} + 4\right)}^{4} + 4 \left(5 {x}^{4} + 0\right) {\left({x}^{3} + 2\right)}^{2} {\left({x}^{5} + 4\right)}^{3}$

$= 6 {x}^{2} \left({x}^{3} + 2\right) {\left({x}^{5} + 4\right)}^{4} + 20 {x}^{4} {\left({x}^{3} + 2\right)}^{2} {\left({x}^{5} + 4\right)}^{3}$

Which, when simplified, is equal to:
$= 2 {x}^{2} \left({x}^{3} + 2\right) {\left({x}^{5} + 4\right)}^{3} \left({13}^{x} 5 + 20 {x}^{2} + 12\right)$