Find the Derivative - d/d@VAR f(x)=8e^x+4 natural log of x^3

Problem

d()/d(x)*(8*ex+4*ln(x3))

Solution

1. Simplify the expression using the logarithm power rule ln(ab)=b*ln(a) to make differentiation easier.

4*ln(x3)=4⋅3*ln(x)=12*ln(x)

2. Apply the sum rule for derivatives, which allows for differentiating each term individually.

d()/d(x)*(8*ex+12*ln(x))=(d(8)*ex)/d(x)+(d(12)*ln(x))/d(x)

3. Differentiate the first term using the rule d(ex)/d(x)=ex

(d(8)*ex)/d(x)=8*ex

4. Differentiate the second term using the rule d(ln(x))/d(x)=1/x

(d(12)*ln(x))/d(x)=12/x

5. Combine the results to find the final derivative.

8*ex+12/x

Final Answer

d()/d(x)*(8*ex+4*ln(x3))=8*ex+12/x

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