Video Transcript
If 𝑓 of 𝑥 is equal to negative
five 𝑒 to the negative nine 𝑥, find 𝑓 dash of 𝑥. Find the derivative of the
function.
Remember, the derivative of 𝑒 to
the power of 𝑥 is 𝑒 to the power of 𝑥. And the derivative of 𝑒 to the
power of 𝑘𝑥 is 𝑘𝑒 to the 𝑘𝑥. Now, our function is some multiple
of 𝑒 to the 𝑘𝑥. It’s negative five times 𝑒 to the
𝑘𝑥, where 𝑘 is equal to negative nine. Now, we know that the constant
factor rule allows us to take constants outside a derivative and concentrate on
differentiating the function of 𝑥 itself. So this means we can say that the
derivative of the function of 𝑥 is equal to negative five times the derivative of
𝑒 to the negative nine 𝑥. And we know that the derivative of
𝑒 to the negative nine 𝑥 is negative nine times 𝑒 to the negative nine 𝑥. And since negative five multiplied
by negative nine is 45, we can say that the derivative of our function is 45𝑒 to
the negative nine 𝑥.
