Use logarithmic differentiation to determine the derivative.

Question

Let f(x)=x^7x. Use logarithmic differentiation to determine the derivative.

F(x)= ?

F(1)=?

Answer 1

y = x^7x

ln(y) = ln(x^7x)

ln(y) = (7x)ln(x)

Now take the derivative of both sides with respect to x. The left side require the use of the chain rule. The right side requires the use of the product rule:

Left side:

[ln(y)]' = (1/y)(dy/dx)

Right side:

[(7x)(ln(x))]' = (7)(ln(x)) + (7x)(1/x) = 7ln(x) + 7

Putting these together:

(1/y)(dy/dx) = 7ln(x) + 7

Multiply both sides by "y":

dy/dx = y(7ln(x) + 7) but "y" = x^7x so:

dy/dx = x^7x(7ln(x) + 7)

dy/dx = 7x^7x(ln(x) + 1)