Calculate the derivative of the following functions.
y =...

Question

Calculate the derivative of the following functions.

y = (f(g(x^m)))^n, where f and g are differentiable for all real numbers and m and n are constants

Accepted Answer

Hi, everyone, let's take a look at this practice problem dealing with derivatives. This problem says to find the derivative of the function below, and we're given the function Y is equal to the quantity of the sign of cosine of x 5th and quantity rates to the 9th. We have 4 possible choices as our answers. Your choice A, we have -45, multiplying the quantity of sine of cosine of x to the 5th. In quantity rates to the 8th, multiplied by the cosine of cosine of x 5th, multiplied by sin of x 5. For choice B, we have minus 45, multiplied by X 4th, multiplying the quantity of sine of cosine of x 5th in quantity raised to the 8th, multiplied by the cosine of cosine of x 5th. And for choice D, we have 45 multiplied by X the 4th, multiplying the quantity of sine of cosine of X 5th in quantity raised to the 8th, multiplying the cosine of cosine of X 5th, multiplied by sine of X 5th. And for choice D, we have -45 multiplied by X the 4th, multiplying the quantity of sine of cosine of X 5th, in quantity raised to the 8th, multiplying the cosine of cosine of X 5th, multiplied by sine of X 5th. So, we need to take the derivative of the function that we were given in the problem, and to take the derivative of this function, we're going to need to use the chain rule. So recall for the chain rule that we have the derivative with respect to X. Of the quantity of U of V of X. And when we take the derivative of this quantity, this is going to be equal to. U, which is the derivative of U with respect to X of quantity of the of X. Multiplied by the derivative of V with respect to X V of X. So we're actually going to use this chain rule multiple times to take the derivative of our function Y here. So, we're gonna be looking for the derivative of Y with respect to X, and this is going to be equal to, when I take the derivative of the quantity that's inside the print C that's rates the 9th power, I'm gonna get 9. Multiplied by that quantity, which is sine of cosine. Of X to the 5th. In quantity And when I've taken the derivative here, I'm going to subtract 1 from the X1, so we'll have 9 x 1, which is 8, so that quantity is raised to the 8th power. We then need to multiply this by the derivative with respect to X. Of the quantity that is inside our parentheses, which is sign of cosine. Of X to the 5th. We now need to take that derivative. In doing so, this is going to be equal to. Our first term remains the same, so we'll have 9 multiplying the quantity of sine of cosine of X 5th. In quantity, raised to the 8th. And when I take the derivative of sine, I'll get cosine. So I'll have cosine. Of, and here we'll just keep the original argument, which is cosine of X to the 5th. And then I'll need to multiply that by the derivative with respect to X of the argument, which in this case was cosine of X to the 5th. So now taking that derivative, we'll get, and so we will have for the first two terms, the same values, so we'll have 9, multiplying the quantity of sign. Of cosine of x to the 5th. In quantity, raised to the 8th power, multiplied by the cosine of cosine of X to the 5th. Then we've taken the derivative of cosine here, which if you recall, is equal to minus signs, we'll have minus sign. Of X to the 5th. And then we have to multiply that by the derivative with respect to X of X-rays to the 5th power. So continuing to take that derivative. Our full derivatives is going to be equal to, we'll keep the first um three terms the same, so we'll have 9 multiplied by the sign of cosine. Of X to the 5th. In quantity, raised to the 8th power, multiplied by cosine. Of cosine of x to the 5th. And then that's gonna be multiplied by. Minus sign Of X to the 5th. And then that's multiplied by the derivative of X to the 5th, which is 5 multiplied by X 4th. So we can just simplify our expression a bit, so this is going to be equal to, we'll bring the minus sign out front, so we'll have minus, we'll have 9 multiplied by 5, which is 45. And then the rest of the terms are the same, so we'll have 45 multiplying of multiplying X to the 4th. Multiplying the quantity of sign. Of cosine. Of X to the 5th. In quantity, raised to the 8th, multiplied by cosine of cosine of X to the 5th. Multiplying Sign of X to the 5th. So that is the answer that we were looking for. And now if we come up to our answer choices. And compare them to the answer that we calculated, the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Calculate the derivative of the following functions.
y = (f(g(x^m)))^n, where f and g are differentiable for all real numbers and m and n are constants

Key Concepts

Chain Rule

Product Rule

Power Rule

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