Chain Rule: The General Exponential Rule - Concept
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.