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

2 Answers
Oct 24, 2015

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)

Oct 24, 2015

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)