Question

find (a) the derivative of $\frac{t(x)}{b(x)}$ without using the quotient rule, and (b) $\frac{t(x)}{b(x)}$. note that the answer to part (b) is different from the answer to part (a).
$t(x)=x^{6}$, $b(x)=x^{4}$
(a) the derivative of $\frac{t(x)}{b(x)}$ is □

Explanation:

Step1: Simplify the function

First, simplify $\frac{T(x)}{B(x)}$. Given $T(x)=x^{6}$ and $B(x)=x^{4}$, then $\frac{T(x)}{B(x)}=\frac{x^{6}}{x^{4}}=x^{6 - 4}=x^{2}$ by the rule $\frac{x^{m}}{x^{n}}=x^{m - n}$.

Step2: Differentiate the simplified function

Use the power - rule for differentiation, which states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$. For $y=x^{2}$, the derivative $y^\prime = 2x^{2-1}=2x$.

Answer:

$2x$