In Exercises 59–86, find the derivative of y with respect to t...

Question

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

65. y = (cos θ)^(√2)

Accepted Answer

Welcome back, everyone. Find D Y divided by DX4 Y equals sin x to the power of three halves. It says 3 halves cosine x, sine x to the power of 1/2, B 3 halves, sine x, cosine x to the power of 1/2, C 1/2 cosine x, sine x to the power of 1/2, and D 12 cosine x, sine x to the power of three halves. For this problem, let's recognize that Y is a composite function. We have an error function, 3 halves is basically our exponent, right, and the inner function is sin x. So we're going to use the U substitution. Let's suppose that u equals our inner function sine x, and then Y can be written as you raise the power of three halves. If we want to differentiate a composite function, we have to apply the chain rule so that DY divided by DX is equal to Dy divided by DU multiplied by. DU divided by DX. That's the chain rule. Let's identify each derivative. The derivative of Y with respect to U is equal to the derivative of U to the power of three halves. We can apply the power rule here, right? So we bring down the exponents, 3 halves, and the original exponent reduces by 1 unit. So 3 halves minus 1 is 1/2. We can now express it in terms of X, so that'd be 3 halves. S X erase the power of one half. And now let's find the derivative of you with respect to X. That's the derivative of sine of X, which is equal to cosine of X, right? Now, if we take these results and substitute into the chain rule, we're going to get 3 halves multiplied by cosine of X. Multiplied by sine of x the power of one half. Which corresponds to the answer choice a. Thank you for watching.

In Exercises 59–86, find the derivative of y with respect to the given independent variable.65. y = (cos θ)^(√2)

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In Exercises 59–86, find the derivative of y with respect to the given independent variable.
65. y = (cos θ)^(√2)