Find f'(x): f(x)=4x^5 ln x

Question

f'(x)=

Answer 1

To find f'(x) we can use the product rule and the chain rule.

Let's take f(x) = u(x)v(x), where u(x) = 4x^5 and v(x) = ln(x).

The product rule states that (u*v)' = u'v + uv'.

Using the power rule, the derivative of u(x) with respect to x is:

u'(x) = d/dx (4x^5) = 20x^4

The derivative of v(x) = ln(x) with respect to x is:

v'(x) = d/dx (ln(x))
v'(x) = 1/x (using the derivative of ln(x))

Applying the product rule, we have:

f'(x) = (u*v)' = u'v + uv'
f'(x) = (20x^4)(ln(x)) + (4x^5)(1/x)

Simplifying further,

f'(x) = 20x^4 ln(x) + 4x^4

Therefore, the derivative of f(x) = 4x^5 ln(x) is f'(x) = 20x^4 ln(x) + 4x^4.