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

Question

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

y = In (x³+1)^π

Accepted Answer

Welcome back, everyone. For the given function, finds the derivative. Y equals Ln of 3 x 5th plus 7 raises to the power of E. We're given 4 answer choices A says 15 E X cubed divided by 3x the power of 5 plus 7, B says 15 X the power of 4th divided by 3x the power of 5th plus 7. C says. 15 e x the power of 4th divided by 3x the power of fifth plus 7 squad and D says -15 X to the power of 4th divided by 3 x 5 plus 7. So we're given our function. Let's rewrite it. It says Y equals ln of 3x the power of 5th plus 7 raise to the power of E. Whenever we want to perform differentiation, the first step is to check if we can simplify the given function. And in this case we can notice that there's an exponent within our logarithm and based on the properties of logarithms we can bring down this exponent in front of the log, right? We can write y equals e multiplied by LN of 3x the power of fifth plus 7, and E is a constant. It's the base of the natural log, right. So we want to evaluate the derivative of Y, we can factor out the constant, so that's just E and E is multiplied by the derivative of LN of 3 X to the power of 5th plus 7. And for this problem, well, we can apply the chain rule. So let's write Y equals e multiplied by. Let's recall that the derivative of L and X is 1 divided by X, right? We're going to say E is multiplied by 1 divided by 3 x to the power of 5 + 7 because this is what we have inside of the log. However, this is not X. It is a function, and this means that we need to apply the chain rule. So we're multiplying by the derivative of the inner function or the derivative of 3x the power of fifth plus 7, and this is our expression that we want to evaluate. So we get e. Divided by 3 x to the power of 5th plus 7th and now let's evaluate the remaining derivative. Here we can apply the power rule so we end up with 3 multiplied by the derivative of X to the power of fifth is 5 x to the power of fourth based on the power rule. We bring down the 5 in the original exponent decreases by 1 unit and the derivative of 7 is 0. So overall we are multiplying by 15. X to the power of 4th because 3 multiplied by 5 is 15. And now if we simplify, we're going to get 15. E X to the power of 4th divided by 3 x to the power of 5th. Plus 7 which corresponds to the answer choice b. Thank you for watching.

15–48. Derivatives Find the derivative of the following functions.y = In (x³+1)^π

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15–48. Derivatives Find the derivative of the following functions.
y = In (x³+1)^π