Find the Derivative - d/dx x^2 natural log of 5x

Problem

d()/d(x)*x2*ln(5*x)

Solution

1. Identify the product rule, which states that for two functions u and v the derivative is ud(v)/d(x)+vd(u)/d(x)

2. Assign the parts of the expression to the variables u=x2 and v=ln(5*x)

3. Differentiate u to find d(u)/d(x)=2*x

4. Differentiate v using the chain rule to find d(ln(5*x))/d(x)=1/(5*x)⋅5=1/x

5. Apply the product rule formula by substituting the parts back in.

x2⋅1/x+ln(5*x)⋅2*x

6. Simplify the resulting expression by performing the multiplication.

x+2*x*ln(5*x)

7. Factor out the common term x to reach the final form.

x*(1+2*ln(5*x))

Final Answer

(d(x2)*ln(5*x))/d(x)=x*(1+2*ln(5*x))

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