Find the derivative of the function.
$$f ( x ) = \left( x ^ { 2 } + 1 \right) \left( \frac { 9 x - 1 } { 7 x + 1 } \right)$$
A) $\frac { 63 x ^ { 3 } + 135 x ^ { 2 } + 49 x + 9 } { 7 ( 7 x + 1 ) ^ { 2 } }$
B) $\frac { 63 x ^ { 3 } + 135 x ^ { 2 } + 49 x + 9 } { 7 ( 7 x + 1 ) }$
C) $\frac { 126 x ^ { 3 } + 20 x ^ { 2 } - 2 x + 16 } { ( 7 x + 1 ) ^ { 2 } }$
D) $\frac { 126 x ^ { 3 } + 20 x ^ { 2 } - 2 x + 16 } { ( 7 x + 1 ) }$

Question

Quizplus · Accepted Answer

Using the product rule, we get:
$$f'(x)=\left(x^2+1\right)\frac{d}{dx}\left(\frac{9x-1}{7x+1}\right)+\frac{9x-1}{7x+1}\frac{d}{dx}\left(x^2+1\right)$$
$$=\left(x^2+1\right)\frac{d}{dx}\left(\frac{9x-1}{7x+1}\right)+2x\frac{9x-1}{7x+1}$$
Now we need to apply the quotient rule to the first term:
$$\frac{d}{dx}\left(\frac{9x-1}{7x+1}\right)=\frac{(9)(7x+1)-(9x-1)(7)}{(7x+1)^2}=\frac{126x^2+20x-2}{(7x+1)^2}$$
Substituting this back into our original expression, we get:
$$f'(x)=\left(x^2+1\right)\cdot\frac{126x^2+20x-2}{(7x+1)^2}+2x\cdot\frac{9x-1}{7x+1}$$
$$=\frac{126 x ^ { 3 } + 20 x ^ { 2 } - 2 x + 16 } { ( 7 x + 1 ) ^ { 2 } }$$

Therefore, the answer is (C).

Find the Derivative of the Function f(x)=(x2+1)(9x−17x+1)f ( x ) = \left( x ^ { 2 } + 1 \right) \left( \frac { 9 x - 1 } { 7 x + 1 } \right)f(x)=(x2+1)(7x+19x−1​)

Find the Derivative of the Function $f ( x ) = \left( x ^ { 2 } + 1 \right) \left( \frac { 9 x - 1 } { 7 x + 1 } \right)$