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Video: Finding the First Derivative of a Function Involving an Exponential Function

Alex Cutbill

Find the first derivative of the function 𝑦 = 4𝑥⁶ + 2𝑒^𝑥.

01:37

Video Transcript

Find the first derivative of the function 𝑦 equals four 𝑥 to the six plus two 𝑒 to the 𝑥.

We’re looking then for 𝑑𝑦 by 𝑑𝑥 that is a derivative with respect to 𝑥 of four 𝑥 to the six plus two 𝑒 to the 𝑥. And as the derivative of the sum of two functions is the sum of their derivatives, we can split up 𝑑 by 𝑑𝑥 of four 𝑥 to the six plus two 𝑒 to the 𝑥 into the two derivatives 𝑑 by 𝑑𝑥 four 𝑥 to the six and 𝑑 by 𝑑𝑥 two 𝑒 to the 𝑥.

We find these two derivatives using different methods. For the first of these derivatives, we can use the fact that the derivative with respect to 𝑥 of 𝑎 times 𝑥 to the 𝑛 is 𝑎 times 𝑛 times 𝑥 to the 𝑛 minus one. The coefficient becomes four times six, which is 24. And the exponent becomes six minus one, which is five.

For the other derivative, we use the special property of the number 𝑒 — that the derivative of 𝑒 to the 𝑥 with respect to 𝑥 is just 𝑒 to the 𝑥. This fact means that the derivative of two 𝑒 to the 𝑥 with respect to 𝑥 is two 𝑒 to the 𝑥. Here, we also use the fact that the derivative of a number times a function is that number times the derivative of the function.

Using these laws, we see that the first derivative of the function 𝑦 equals four 𝑥 to the six plus two 𝑒 to the 𝑥 is 24𝑥 to the five plus two 𝑒 to the 𝑥.