Find the Derivative - d/dx y=x^(6cos(x))

Problem

d()/d(x)*x(6*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(6*cos(x)) to simplify the exponent.

ln(y)=ln(x(6*cos(x)))

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

ln(y)=6*cos(x)*ln(x)

4. Differentiate implicitly with respect to x on both sides of the equation.

d(ln(y))/d(x)=(d(6)*cos(x)*ln(x))/d(x)

5. Apply the chain rule to the left side and the product rule to the right side.

1/yd(y)/d(x)=(d(6)*cos(x))/d(x)*ln(x)+6*cos(x)d(ln(x))/d(x)

6. Calculate the derivatives of the individual trigonometric and logarithmic terms.

1/yd(y)/d(x)=−6*sin(x)*ln(x)+(6*cos(x))/x

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

d(y)/d(x)=y*(−6*sin(x)*ln(x)+(6*cos(x))/x)

8. Substitute the original expression for y back into the equation to get the final result.

d(y)/d(x)=x(6*cos(x))*((6*cos(x))/x−6*sin(x)*ln(x))

Final Answer

d(x(6*cos(x)))/d(x)=x(6*cos(x))*((6*cos(x))/x−6*sin(x)*ln(x))

Want more problems? Check here!