Question

find (a) the derivative of f(x)s(x) without using the product rule, and (b) f(x)s(x). note that the answer to part (b) is different from the answer to part (a).
f(x)=x^2 + 1, s(x)=x^5
(a) the derivative of f(x)s(x) is

Explanation:

Step1: Multiply the functions

$F(x)S(x)=(x^{2}+1)x^{5}=x^{7}+x^{5}$

Step2: Apply power - rule for derivatives

The power - rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$.
For $y=x^{7}+x^{5}$, the derivative $y^\prime=\frac{d}{dx}(x^{7}+x^{5})=\frac{d}{dx}(x^{7})+\frac{d}{dx}(x^{5})$.
$y^\prime = 7x^{6}+5x^{4}$

Step3: Find $F^\prime(x)$ and $S^\prime(x)$

For $F(x)=x^{2}+1$, using the power - rule, $F^\prime(x)=\frac{d}{dx}(x^{2}+1)=2x$.
For $S(x)=x^{5}$, using the power - rule, $S^\prime(x)=\frac{d}{dx}(x^{5}) = 5x^{4}$.

Step4: Calculate $F^\prime(x)S^\prime(x)$

$F^\prime(x)S^\prime(x)=(2x)\times(5x^{4})=10x^{5}$

Answer:

(A) $7x^{6}+5x^{4}$
(B) $10x^{5}$