Find the derivative of the function

Question

Answer 1

Hello J.M.

For this question, recall the following:

The derivative of e^x is just e^x (d/dx(e^x) = e^x)

However, since we have more than just one operation acting upon 'x' per exponential function listed:

e^-x -> an exponential AND a negative are acting upon 'x' (I am leaving out the '8' because constants have no weight)

e^3x -> an exponential AND a product of 3 are acting upon 'x'.

It looks like your best bet is using chain rule to respect every operation that's acting upon 'x'.

Recall that chain rule is: d/dx(f(g(x))) = f'(g(x))*g'(x) where f and g are operations acting upon 'x'

For e^-x, the outer most operation will be the exponential and the inner most operation is the product of -1.

Moving forward with chain rule:

d/dx(8e^-x) = (8e^-x)*d/dx(-x) where f'(g(x)) = 8e^-x since the derivative of the exponential is just itself, and g'(x) = d/dx(-x) = -1. Recall that the constant 8 in the front does not hold any weight still thus I can factor it out and just deal with the operations that directly involve 'x'.

Thus,

d/dx(e^-x) = -8e^-x.

I will leave the second half e^3x to you.

Hope this helps!