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Chain Rule: The General Exponential Rule - Concept 9,938 views

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

The exponential rule is a special case of the chain rule. It is useful when finding the derivative of e raised to the power of a function. The exponential rule states that this derivative is e to the power of the function times the derivative of the function.

I want to talk about a special case of the chain rule where the function that we're differentiating has its outside function e to the x so in the next few problems we're going to have functions of this type which I call general exponential functions. we'll have e to the x as our outside function and some other function g of x as the inside function.
And I'll have a special version of the chain rule that I'll use for these and I'll call this rule the general exponential rule. So the derivative of e to the g of x is e to the g of x times g prime of x. And this is because the derivative of e to the x if you'll recall derivative of e to the x is just e to the x. Okay let's try this out on h of x equals e to the x squared plus 3x+1 and let's observe that again the outside function is e to the x and the inside function is this polynomial x squared plus 3x+1 and so the derivative according to this formula is the same function e to the g of x right so e to the x squared plus 3x+1 times g prime of x and that's the derivative of the inside function.
And that derivative is 2x+3 and that's it, these are super easy to differentiate so every time you a function of the form e to the g of x it's derivative is e to the g of x times the derivative of the inside function.