Question

How do you use the Product Rule to find the derivative of `2x^4 * 6^(3x)`?

Answer 1

`d/d(2x^4*6^(3x))=(6x^4)(6^(3x)ln6)+6^(3x)(8x^3)`

Explanation:

The product rule states that

`d/dx[f(x)*g(x)]=f(x)*g'(x)+g(x)*f'(x)`

In this case we may take our `f(x)=2x^4 and g(x)=6^(3x)` and apply the product rule together with other standard rules of differentiation to get

`d/d(2x^4*6^(3x))=(2x^4)(6^(3x)ln6*3)+6^(3x)(8x^3)`

Answer 2

See the explanation section, below.

Explanation:

Because it involves two commutative operations (multiplication and addition), the product rule can be written in several orders.

I will use the product rule written `d/dx(uv) = u'v+uv'` (where the prime indicates a derivative with respect to `x`.

`y = 2x^4*6^(3x)`

`u = 2x^4` so `u' = 8x^3` and

`v = 6^(3x)`, so `v' = 6^(3x) ln6 d/dx(3x) = 3*6^(3x) ln6`

`y' = (8x^3)(6^(3x))+(2x^4)(3*6^(3x) ln6)`

Rewrite to taste using algebra. I like;

`y' = 8x^3 6^(3x) + 6x^4 6^(3x) ln6)`

or (better yet)

`y' = 2x^3 6^(3x)(4 + 3x ln6)`