NEED ANSWERED ASAP! Derivatives of exponential functions...

Question

determine the derivative of the following: 

 

g(t)= e^2t / 1+e^2t

Answer 1

Remember that the derivative of e^x = e^x. We're going to need to use the chain rule and the quotient rule for this.

 

 

Chain Rule:

 

To find the derivative of e^2t, let 2t = x. Then the derivative of e^2t is the derivative of e^x (which is e^x, or e^2t) times the derivative of x (which is the derivative of 2t, or 2). The derivative of e^2t is 2e^2t.

 

 

Quotient Rule:

 

g'(t) = ((denominator times derivative of numerator) - (numerator times derivative of denominator)) / (denominator)²

g'(t) = ((1+e^2t)(2e^2t) - (e^2t)(2e^2t)) / (1+e^2t)²

g'(t) = (2e^2t + 2e^2te^2t - 2e^2te^2t) / (1 + e^2t + e^2t + (e^2t)²)

g'(t) = (2e^2t) / (1 + 2e^2t + e^4t)