In Exercises 25–30, use logarithmic differentiation to find th...

Question

In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

29. y = (sin θ)^√θ

Accepted Answer

Welcome back everyone. Use logarithmic differentiation to compute dyy divided by dx for y equals cosine of x raised to the power of x 2. For this problem, let's remember that in logarithmic differentiation we first of all take ln of both sides. So in this case, considering our equation, ln of y on the left hand side is going to be equal to ln of cosine of. X Raise to the power of X2, right? And now what we can do is simply apply the properties of logarithms to show that LN of Y is equal to x2 multiplied by LN of cosine x, right? Using the power rule for logarithms, we can bring down the exponent. From here, we can differentiate both sides. We can show that the derivative of the left-hand side is equal to the derivative of the right-hand side. Starting with the left-hand side, we have a Composite function because y is a function of x, so we have to apply the chain rule. The derivative of l and of y is going to be 1 divided by y. Multiplied by the derivative of y, we can call it y. Based on the fact that y is a function of x, we want to include that term on the right hand side. We want to apply the product rule, so that would be x 2 derivative multiplied by ln of cosine x plus x2 multiplied by ln of cosine x derivative. Let's go ahead and simplify. On the left hand side we have y. Divided by y and we already know that y is cosine of x raises the power of x squares we can substitute. On the right hand side, the derivative of x 2 is 2 x. Which is multiplied by ln of cosine x and we're adding x squared. Multiplied by ln of cosine x derivative. So once again we want to apply the chain rule. We get 1 divided by cosine of x multiplied by the derivative of the inner function, so the derivative of cosine x. And the derivative of cosine x is negative sine x. So now when we combine. Negative sign divided by cosine is negative tangent, so we can just write negative tangent X. And replace the positive sign with the negative sign, so we get x 2 tangent x. Well then, our final step is to solve for y, and we can do that by multiplying both sides by cosine x to the power of x2. So we get y equals cosine x to the power of x2 multiplied by 2 x ln of cosine x minus x2. Tangent x additionally, to perform a complete factorization, we can take out x. Which leaves us with x. Multiplied by cosine x. To the power of x 2 multiplied by 2 ln of cosine x. Minus x tangent x and that will be our final answer for this problem. Thank you for watching.

In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.29. y = (sin θ)^√θ

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In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
29. y = (sin θ)^√θ