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^4 + 1, s(x)=x^6. (a) the derivative of f(x)s(x) is 10x^9 + 6x^5. (b) f(x)s(x)=□

Explanation:

Step1: Find the derivative of F(x)

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, for $F(x)=x^4 + 1$, $F'(x)=\frac{d}{dx}(x^4+1)=4x^{3}$.

Step2: Find the derivative of S(x)

For $S(x)=x^6$, using the power - rule, $S'(x)=\frac{d}{dx}(x^6)=6x^{5}$.

Step3: Calculate $F'(x)S'(x)$

Multiply $F'(x)$ and $S'(x)$: $F'(x)S'(x)=(4x^{3})\times(6x^{5})$.
Using the rule of exponents $a^m\times a^n=a^{m + n}$, we have $F'(x)S'(x)=24x^{3 + 5}=24x^{8}$.

Answer:

$24x^{8}$