Help! How would I answer this chain rule question?

Question

Let f(x) = (In x)⁴

what is:

a) f’(x)= ?

b) f’(e⁴)= ?

Answer 1

Apply the power rule to get to f'(x) = 4(lnx)³, but then multiply by derivative of "inside" to get the final derivative:

f'(x) = 4[ln{x}]³ (1/x)

See the following graph to get some ideas on using Desmos to verify.

desmos.com/calculator/tcmuwjoeem

The green line in the above graph is the tangent at (e⁴,f(e⁴)., You will have to scroll to see the point of tangency. The purple point and line are movable and show other tangent lines for values of x.