*h* is a product of the two functions *f*(*x*) = *x*^{2} and *g*(*x*) = *e*^{x}.
Then *f'*(*x*) = 2*x* and *g'*(*x*) = *e*^{x},
therefore While the answer 2*xe*^{x} + 2*e*^{x} is correct, we factored out 2*e*^{x} to find the answer 2*e*^{x}(*x* + 1) because in later problems, it will be helpful to have the answer in this factored form. While the product rule is usually written (*fg*)' = *f'g* + *fg'*, the letters "f" and "g" are placeholders. To find the derivative of a product of two functions, find what each function multiplied by the derivative of the other is, and add the results. If we avoid the letters *f* and *g*, the product rule says: to find the derivative of ((first function) × (second function)), find (derivative of first function) × second function and add first function × (derivative of second function). We only need to use the product rule when there are two expressions with *x* multiplied together. |