Find the Derivative - d/dx y=x^(8cos(x))

Problem

d()/d(x)*x(8*cos(x))

Solution

1. Identify the function as a variable base raised to a variable exponent, which requires logarithmic differentiation.

2. Apply the natural logarithm to both sides of the equation y=x(8*cos(x)) to simplify the exponent.

ln(y)=ln(x(8*cos(x)))

3. Use the power rule for logarithms to move the exponent in front of the natural log.

ln(y)=8*cos(x)*ln(x)

4. Differentiate implicitly both sides with respect to x using the product rule on the right side.

1/yd(y)/d(x)=(d(8)*cos(x))/d(x)*ln(x)+8*cos(x)d(ln(x))/d(x)

5. Calculate the derivatives of the individual components.

1/yd(y)/d(x)=−8*sin(x)*ln(x)+(8*cos(x))/x

6. Solve for the derivative by multiplying both sides by y

d(y)/d(x)=y*(−8*sin(x)*ln(x)+(8*cos(x))/x)

7. Substitute the original expression for y back into the equation.

d(y)/d(x)=x(8*cos(x))*(−8*sin(x)*ln(x)+(8*cos(x))/x)

Final Answer

d(x(8*cos(x)))/d(x)=x(8*cos(x))*((8*cos(x))/x−8*sin(x)*ln(x))

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