Find the Derivative - d/dx y=cos(x)^x

Problem

d()/d(x)*cos(x)

Solution

1. Rewrite the function using the natural base e and the natural logarithm to handle the variable in the exponent.

y=e^ln(cos(x))

y=e^(x*ln(cos(x)))

2. Apply the chain rule to differentiate the exponential function, where the derivative of e^u is e^u⋅d(u)/d(x)

d(y)/d(x)=e^(x*ln(cos(x)))⋅d()/d(x)*(x*ln(cos(x)))

3. Apply the product rule to the expression x*ln(cos(x)) which states d()/d(x)*(u*v)=u^′*v+u*v^′

d()/d(x)*(x*ln(cos(x)))=ln(cos(x))⋅d(x)/d(x)+x⋅d(ln(cos(x)))/d(x)

4. Apply the chain rule to differentiate ln(cos(x)) knowing the derivative of ln(u) is 1/u⋅d(u)/d(x)

d(ln(cos(x)))/d(x)=1/cos(x)⋅(−sin(x))

d(ln(cos(x)))/d(x)=−tan(x)

5. Substitute the results back into the product rule expression.

d()/d(x)*(x*ln(cos(x)))=ln(cos(x))−x*tan(x)

6. Combine all parts and substitute the original expression for e^(x*ln(cos(x))) to find the final derivative.

d(y)/d(x)=cos(x)*(ln(cos(x))−x*tan(x))

Final Answer

d(cos(x))/d(x)=cos(x)*(ln(cos(x))−x*tan(x))

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