# Video: Differentiating Exponential Functions Using the Chain Rule

If 𝑓(𝑥) = −5𝑒^(−9𝑥), find 𝑓′(𝑥).

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### 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.