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 $2x$.
(b) $\frac{t(x)}{b(x)}=square$

Explanation:

Step1: Find derivatives of T(x) and B(x)

Given $T(x)=x^{6}$, by the power - rule $(x^n)' = nx^{n - 1}$, $T'(x)=6x^{5}$. Given $B(x)=x^{4}$, then $B'(x)=4x^{3}$.

Step2: Calculate $\frac{T'(x)}{B'(x)}$

$\frac{T'(x)}{B'(x)}=\frac{6x^{5}}{4x^{3}}$.

Step3: Simplify the fraction

Using the rule $\frac{x^{m}}{x^{n}}=x^{m - n}$, we have $\frac{6x^{5}}{4x^{3}}=\frac{6}{4}x^{5 - 3}=\frac{3}{2}x^{2}$.

Answer:

$\frac{3}{2}x^{2}$