In Exercises 115–126, use logarithmic differentiation or the m...

Question

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.

120. y = x^(sin x)

Accepted Answer

Welcome back everyone. Use logarithmic differentiation to find the derivative of y with respect to x or y equals x to the power of cosine x. In this problem, the first step is to take ln of both sides. So we're going to get ln of y equals ln of x to the power of cosine x, since we are using logarithmic differentiation, and this allows us to bring down the exponent right using the power rule of logarithms. So that ln of y is equal to cosine x multiplied by ln of x. And for this new equation, we can basically differentiate both sides. On the left hand side, we're going to take. The derivative of l and of y, which is 1 divided by y, multiplied by y derivative using the chain rule because y is a function of x. And on the right hand side we want to apply the product rule since we have the derivative of the product. Applying the product rule, we take cosine. X derivative multiplied by L of x. Plus cosine x multiplied by eon of x derivative. On the left-hand side, we have Y divided by y, and on the right-hand side, the derivative of cosine is negative sin x. Which is now multiplied by eon of x, and we're adding cosine x multiplied by eon of x derivative, which is 1 divided by x. From here, let's go ahead and simplify. We can first of all write. Our positive term so that y divided by y is going to be equal to. Cosine x divided by x minus Sign x Ln. X Now, let's multiply both sides by y since we're solving for Y, meaning Y equals. Y multiplied by Cosine x divided by x minus sin x ln x. We can also factor out 1 divided by x. To ensure that we don't have any fractions in parentheses, which gives us y equals 1 divided by x multiplied by y multiplied by cosine x. Minus xinx ln x, right? And now the actual y is given, it is x to the power of cosine x. The final step is to simplify even further. We can now show that the derivative of y is x, the power of cosine x minus 1, because we have x cosine x divided by x, so we can apply the properties of exponents. We're dividing two terms with the same base, but the different exponents, so we're just subtracting the exponents, cosine x minus 1, right? And therefore y equals x to the power of cosine x minus 1 multiplied by cosine x minus x sin x lnx, and that's our final answer. Thank you for watching.

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.120. y = x^(sin x)

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In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
120. y = x^(sin x)