Find the Derivative - d/d@VAR f(x)=(3x-8) natural log of 2x^5+3

Problem

d()/d(x)*((3*x−8)*ln(2*x5+3))

Solution

1. Identify the rule needed for the derivative, which is the product rule d()/d(x)*(u⋅v)=u⋅d(v)/d(x)+v⋅d(u)/d(x) where u=3*x−8 and v=ln(2*x5+3)

2. Differentiate the first part u=3*x−8 with respect to x

d(3*x−8)/d(x)=3

3. Differentiate the second part v=ln(2*x5+3) using the chain rule, which states d(ln(g(x)))/d(x)=1/g(x)⋅d(g(x))/d(x)

d(ln(2*x5+3))/d(x)=1/(2*x5+3)⋅10*x4

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

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

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

5. Simplify the expression by distributing and rearranging the terms.

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

d(ƒ)/d(x)=(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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