find (a) the derivative of $\frac{t(x)}{b(x)}$ without using the quotient rule, and (b) $\frac{t(x)}{b(x)}$…

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).\n$t(x)=x^{7}, b(x)=x^{2}$\n(a) the derivative of $\frac{t(x)}{b(x)}$ is $5x^{4}$.\n(b) $\frac{t(x)}{b(x)}=square$

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).\n$t(x)=x^{7}, b(x)=x^{2}$\n(a) the derivative of $\frac{t(x)}{b(x)}$ is $5x^{4}$.\n(b) $\frac{t(x)}{b(x)}=square$

Answer

Explanation:

Step1: Find $T'(x)$ and $B'(x)$

Using the power - rule $(x^n)'=nx^{n - 1}$, for $T(x)=x^7$, $T'(x)=7x^{6}$; for $B(x)=x^2$, $B'(x)=2x$.

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

Substitute $T'(x)$ and $B'(x)$ into the fraction: $\frac{T'(x)}{B'(x)}=\frac{7x^{6}}{2x}$.

Step3: Simplify the fraction

Using the rule of exponents $\frac{x^m}{x^n}=x^{m - n}$, we have $\frac{7x^{6}}{2x}=\frac{7}{2}x^{6 - 1}=\frac{7}{2}x^{5}$.

Answer:

$\frac{7}{2}x^{5}$