Find the Derivative - d/dx 2x natural log of 8x+x

Problem

d()/d(x)*2*x*ln(8*x+x)

Solution

1. Simplify the expression inside the natural logarithm by combining the like terms 8*x and x

2*x*ln(8*x+x)=2*x*ln(9*x)

2. Identify the product rule, which states that d()/d(x)*ƒ(x)*g(x)=ƒ(x)′*g(x)+ƒ(x)*g(x)′ where ƒ(x)=2*x and g(x)=ln(9*x)

(d(2)*x*ln(9*x))/d(x)=(d(2)*x)/d(x)*ln(9*x)+2*xd(ln(9*x))/d(x)

3. Differentiate the first part of the product, 2*x using the power rule.

(d(2)*x)/d(x)=2

4. Differentiate the second part of the product, ln(9*x) using the chain rule.

d(ln(9*x))/d(x)=1/(9*x)⋅9

d(ln(9*x))/d(x)=1/x

5. Substitute the derivatives back into the product rule formula.

(d(2)*x*ln(9*x))/d(x)=2*ln(9*x)+2*x⋅1/x

6. Simplify the resulting expression by canceling x in the second term.

(d(2)*x*ln(9*x))/d(x)=2*ln(9*x)+2

Final Answer

d()/d(x)*2*x*ln(8*x+x)=2*ln(9*x)+2

Want more problems? Check here!