Find the 2nd Derivative f(x)=e^(7x^2+3)

Problem

d2()/(d(x)2)*e(7*x2+3)

Solution

1. Identify the function ƒ(x)=e(7*x2+3) and the need to apply the chain rule for the first derivative.

2. Apply the chain rule to find the first derivative ƒ(x)′ by multiplying the derivative of the exponent by the original exponential function.

ƒ(x)′=e(7*x2+3)⋅d(7*x2+3)/d(x)

ƒ(x)′=14*x*e(7*x2+3)

3. Apply the product rule to find the second derivative ƒ(x)″ where the two functions are u=14*x and v=e(7*x2+3)

ƒ(x)″=d(14*x)/d(x)⋅e(7*x2+3)+14*x⋅d(e(7*x2+3))/d(x)

4. Differentiate each part using the power rule and the chain rule result from step 2.

ƒ(x)″=14⋅e(7*x2+3)+14*x⋅(14*x*e(7*x2+3))

5. Simplify the expression by performing the multiplication and factoring out the common exponential term.

ƒ(x)″=14*e(7*x2+3)+196*x2*e(7*x2+3)

ƒ(x)″=14*e(7*x2+3)*(1+14*x2)

Final Answer

d2()/(d(x)2)*e(7*x2+3)=14*e(7*x2+3)*(14*x2+1)

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