Find dy/dx y=(2x+1)^(4x)

Problem

y=(2*x+1)(4*x)

Solution

1. Apply the natural logarithm to both sides of the equation to handle the variable in the exponent.

ln(y)=ln((2*x+1)(4*x))

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

ln(y)=4*x*ln(2*x+1)

3. Differentiate implicitly with respect to x on both sides.

d(ln(y))/d(x)=(d(4)*x*ln(2*x+1))/d(x)

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

1/yd(y)/d(x)=(d(4)*x)/d(x)*ln(2*x+1)+4*xd(ln(2*x+1))/d(x)

5. Compute the derivatives of the individual terms on the right side.

1/yd(y)/d(x)=4*ln(2*x+1)+4*x2/(2*x+1)

6. Simplify the expression on the right side.

1/yd(y)/d(x)=4*ln(2*x+1)+(8*x)/(2*x+1)

7. Solve for dy/dx by multiplying both sides by y

d(y)/d(x)=y*(4*ln(2*x+1)+(8*x)/(2*x+1))

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

d(y)/d(x)=(2*x+1)(4*x)*(4*ln(2*x+1)+(8*x)/(2*x+1))

Final Answer

d(y)/d(x)=(2*x+1)(4*x)*(4*ln(2*x+1)+(8*x)/(2*x+1))

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