Find the Derivative - d/dx (3x-8) natural log of 2x^5+3

Problem

d()/d(x)*(3*x−8)*ln(2*x5+3)

Solution

1. Identify the product rule, which states that d()/d(x)*ƒ(x)*g(x)=ƒ(x)d(g(x))/d(x)+g(x)d(ƒ(x))/d(x)

2. Assign the functions ƒ(x)=3*x−8 and g(x)=ln(2*x5+3)

3. Differentiate ƒ(x) to find d(3*x−8)/d(x)=3

4. Differentiate g(x) using the chain rule, where d(ln(u))/d(x)=1/ud(u)/d(x)

5. Calculate the derivative of the inner function d(2*x5+3)/d(x)=10*x4

6. Combine the chain rule components to find d(ln(2*x5+3))/d(x)=(10*x4)/(2*x5+3)

7. Apply the product rule formula by substituting the parts.

d()/d(x)*(3*x−8)*ln(2*x5+3)=(3*x−8)(10*x4)/(2*x5+3)+ln(2*x5+3)*(3)

8. Simplify the expression by distributing and rearranging terms.

d()/d(x)*(3*x−8)*ln(2*x5+3)=(30*x5−80*x4)/(2*x5+3)+3*ln(2*x5+3)

Final Answer

d()/d(x)*(3*x−8)*ln(2*x5+3)=(30*x5−80*x4)/(2*x5+3)+3*ln(2*x5+3)

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