Find the Derivative - d/d@VAR f(x)=(e^(7x^2-3))/( natural log of 3x+5)

Problem

d()/d(x)(e(7*x2−3))/ln(3*x+5)

Solution

1. Identify the rule needed for the derivative of a quotient, which is d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator u=e(7*x2−3) and the denominator v=ln(3*x+5)

3. Differentiate the numerator u using the chain rule, where d(eg)/d(x)=eg⋅d(g)/d(x)

d(e(7*x2−3))/d(x)=e(7*x2−3)⋅14*x

4. Differentiate the denominator v using the chain rule, where d(ln(g))/d(x)=1/g⋅d(g)/d(x)

d(ln(3*x+5))/d(x)=3/(3*x+5)

5. Substitute these components into the quotient rule formula.

d(ƒ(x))/d(x)=(ln(3*x+5)⋅14*x*e(7*x2−3)−e(7*x2−3)⋅3/(3*x+5))/((ln(3*x+5))2)

6. Simplify the expression by factoring out the common term e(7*x2−3) from the numerator.

d(ƒ(x))/d(x)=(e(7*x2−3)*(14*x*ln(3*x+5)−3/(3*x+5)))/((ln(3*x+5))2)

Final Answer

d()/d(x)(e(7*x2−3))/ln(3*x+5)=(e(7*x2−3)*(14*x*ln(3*x+5)−3/(3*x+5)))/((ln(3*x+5))2)

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