In Exercises 1-20 , find the derivative. y=2^x^3

Name: Problem 3, Chapter 3: Calculus for AP
Uploaded: 2020-10-24T08:30:52-08:00
Duration: 5 min 7 s
Channel: Sophie S
Description: In Exercises 1-20 , find the derivative. y=2^x^3

Key Concepts

Exponential Differentiation Rule

This rule involves differentiating functions of the form a^(g(x)), where a is a constant and g(x) is a function of x. The derivative is given by a^(g(x)) multiplied by the natural logarithm of the base a and by the derivative of the exponent function g(x). This formula is essential when dealing with exponentials that have non-simple, variable exponents.

Chain Rule

The chain rule is a fundamental technique in calculus for differentiating composite functions. It states that the derivative of a composite function f(g(x)) is the derivative of the outer function evaluated at g(x) multiplied by the derivative of the inner function g(x). This rule is crucial when the argument of an exponential function is itself a function of x.

Transcript

00:01 So in this problem, we're asked to solve for the derivative of this function y, where y is equal to 2 raise to x cubed.

00:10 So what we're first going to do is we're going to introduce the rules that we need in order to solve this derivative.

00:15 So the first of those is going to be the chain rule, which we can apply when we have a composite function, which is basically just a function inside another function.

00:24 So as we can see here, if a composite function takes the form f of g of x, we have our inner function g of x and our outer function f of x.

00:37 So if we look at our original function y, we can see that f of x would be equal to two raised to the x and g of x would be x cubed.

01:04 And now in order to solve for the derivative of this composite function, what we're going to do is apply this rule.

01:13 So we'll have the derivative of f of x with the inner function intact.

01:23 And then we'll have that multiplied by the derivative of g of x.

01:30 So this is how we're going to apply the chain rule in order to derive the composite function.

01:37 Now another thing that we need to do is to be able to derive the function in the form be raised to the x.

01:43 So when we have a constant raised to our variable x.

01:49 So if this was our function, how would we go about and derive that? well, we can go ahead and apply this rule for deriving functions of this form...

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