Question

find (a) the derivative of $\frac{t(x)}{b(x)}$ without using the quotient rule, and (b) $\frac{t^{prime}(x)}{b^{prime}(x)}$. note that the answer to part (b) is different from the answer to part (a).
$t(x)=x^{7}, b(x)=x^{2}$
(a) the derivative of $\frac{t(x)}{b(x)}$ is $\square$.

Explanation:

Step1: Simplify the function

First, simplify $\frac{T(x)}{B(x)}$. Given $T(x)=x^{7}$ and $B(x)=x^{2}$, then $\frac{T(x)}{B(x)}=\frac{x^{7}}{x^{2}}=x^{7 - 2}=x^{5}$ according to the rule $\frac{x^{m}}{x^{n}}=x^{m - n}$.

Step2: Differentiate the simplified function

The derivative of $y = x^{n}$ is $y'=nx^{n - 1}$. For $y=x^{5}$, its derivative is $y' = 5x^{4}$.

Answer:

$5x^{4}$