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

Problem

d2()/(d(x)2)*(3*x3+7)7

Solution

1. Apply the power rule and chain rule to find the first derivative ƒ(x)′

ƒ(x)′=7*(3*x3+7)6⋅d(3*x3+7)/d(x)

2. Differentiate the inner function 3*x3+7 to complete the first derivative.

ƒ(x)′=7*(3*x3+7)6⋅9*x2

3. Simplify the expression for ƒ(x)′ by multiplying the constants.

ƒ(x)′=63*x2*(3*x3+7)6

4. Apply the product rule to find the second derivative ƒ(x)″ where the two functions are 63*x2 and (3*x3+7)6

ƒ(x)″=(d(63)*x2)/d(x)⋅(3*x3+7)6+63*x2⋅d(3*x3+7)/d(x)

5. Differentiate each part using the power rule and the chain rule for the second term.

ƒ(x)″=126*x*(3*x3+7)6+63*x2⋅6*(3*x3+7)5⋅9*x2

6. Simplify the second term by multiplying the constants and variables.

ƒ(x)″=126*x*(3*x3+7)6+3402*x4*(3*x3+7)5

7. Factor out the greatest common factor, which is 126*x*(3*x3+7)5

ƒ(x)″=126*x*(3*x3+7)5*((3*x3+7)+27*x3)

8. Combine like terms inside the parentheses.

ƒ(x)″=126*x*(3*x3+7)5*(30*x3+7)

Final Answer

d2()/(d(x)2)*(3*x3+7)7=126*x*(3*x3+7)5*(30*x3+7)

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