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