Derivatives with logs

Subject: Math-Derivatives
Name: kate

Question:
What is the derivative of 5 to the 5x-2 at x equals 0.8?

Hi Kate,

Suppose that you call your function y, then

y = 5^{5x - 2}

To differentiate y first take the natural log of both sides and use the fact that log(a^b) = b log(a). Thus

log(y) = log(5^{5x - 2}) = (5x - 2) log(5)

Now differentiate both sides with respect to x to get

(1/y) dy/dx = 5 log(5) or dy/dx = 5y log(5)

To complete the problem, substitute y = 5^{5x - 2} and evaluate when x = 0.8.

Cheers,
Harley