Find the Derivative - d/dx y=(3x+7) natural log of 2x-1

Problem

d()/d(x)*(3*x+7)*ln(2*x−1)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, u=3*x+7 and v=ln(2*x−1) use the product rule: d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

2. Differentiate the first function u=3*x+7 with respect to x

d(3*x+7)/d(x)=3

3. Differentiate the second function v=ln(2*x−1) using the chain rule.

d(ln(2*x−1))/d(x)=1/(2*x−1)⋅d(2*x−1)/d(x)

d(ln(2*x−1))/d(x)=2/(2*x−1)

4. Apply the product rule by substituting the derivatives back into the formula.

d(y)/d(x)=(3*x+7)⋅2/(2*x−1)+ln(2*x−1)⋅3

5. Simplify the expression by combining terms.

d(y)/d(x)=(2*(3*x+7))/(2*x−1)+3*ln(2*x−1)

d(y)/d(x)=(6*x+14)/(2*x−1)+3*ln(2*x−1)

Final Answer

d()/d(x)*(3*x+7)*ln(2*x−1)=(6*x+14)/(2*x−1)+3*ln(2*x−1)

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