`y = (cos(x))^x` Use logarithmic differentiation to find the derivative of the function.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Differentiate `y=(cos(x))^x ` :

Take the natural logarithm of both sides:

`lny=ln(cos(x))^x `

Use the power property of logarithms:

`lny=xln(cos(x)) `

Differentiate; use the product rule on the RHS:

` (dy)/(dx)(1/y)=ln(cos(x))+x(-sin(x))/(cos(x)) `

`y'=y(ln(cos(x))-xtan(x)) `

Substituting for y we get:

` y'=(cos(x))^x(ln(cos(x))-xtan(x)) `

Unlock
This Answer Now

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Start your 48-Hour Free Trial

Differentiate `y=(cos(x))^x ` :

Take the natural logarithm of both sides:

`lny=ln(cos(x))^x `

Use the power property of logarithms:

`lny=xln(cos(x)) `

Differentiate; use the product rule on the RHS:

` (dy)/(dx)(1/y)=ln(cos(x))+x(-sin(x))/(cos(x)) `

`y'=y(ln(cos(x))-xtan(x)) `

Substituting for y we get:

` y'=(cos(x))^x(ln(cos(x))-xtan(x)) `

Approved by eNotes Editorial Team