15–48. Derivatives Find the derivative of the following functi...

Question

15–48. Derivatives Find the derivative of the following functions.

y = 5^3t

Accepted Answer

Welcome back everyone. For the given function, find the derivative. Y equals 31, raise to the power of 2 X. We're given 4 answer choices. A says 2x multiplied by 31 of 2x minus 1, B2 multiplied by 31 of 2 X. C. 2 multiplied by 31, raise the power of 2 X multiplied by LN of 31. And D2 multiplied by 31 to the power of 2 X divided by LN of 31. Let's notice that the given function Y corresponds to a form of A, where A is the number, raise to the power of F of X, right? We can recall that the derivative of a the power of X is equal to a to the power of x multiplied by LN A. So now if our exponent is not X, but it is a function, we simply want to apply the Chain rule. So in this case we have a raise to the power of F of X, and its derivative is going to be equal to a to the power of F of X multiplied by LN A. So it's quite similar, right, but because we have a function in our exponent, we have to apply the chain rule. So we multiply it by F. In this case, we can apply the given formula and we can say that the derivative of 31. Raise the power of 2 X is going to be A, A is 31, raised to the original exponent, which is 2 X. We multiply that by LN of A, which is LN of 31. And then because our exponent is not an independent variable, but it is a function, we have to apply the chain rule. So we multiply by the derivative of 2 X. So the final derivative that we want to find is the derivative of 2 X, which is identical to 2 multiplied by the derivative of X, and the derivative of X based on the power rule is 1. So overall, the derivative of 2 X is 2. And we can say that the final answer is 2 multiplied by 31, raise to the power of 2 XLN of 31, which corresponds to the answer choice C. Thank you for watching.

15–48. Derivatives Find the derivative of the following functions.
y = 5^3t

Key Concepts

Derivatives

Exponential Functions

Power Rule

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