Check the solution to "Determine the derivative of the following functions. f(x)=2x3x2+1"

Clifland

Answered question

2021-10-22

Determine the derivative of the following functions.

f (x) = \frac{2 x}{3 x^{2} + 1}

Answer & Explanation

un4t5o4v

Skilled2021-10-23Added 105 answers

Step 1
The product rule of differentiation or the quotient rule can be used to differentiate a given function. Product rule states that: (fg)'=f'g+fg'. For the function $\frac{f}{g}$ to use product rule write the function as $f g^{- 1}$ .
The quotient rule for differentiation states that ${(\frac{f}{g})}^{'} = \frac{f^{'} g - f g^{'}}{g^{2}}$ . The derivative of $x^{n}, n \neq 0$ is equal to $n x^{n - 1}$ . Use the property of derivatives $(f (x) + g (x))^{'} = f^{'} (x) + g^{'} (x)$ and $(c f (x))^{'} = c f^{'} (x)$ .
Step 2
Given function is $f (x) = \frac{2 x}{3 x^{2} + 1}$ . Use quotient rule of differentiation to find the derivative.
$f (x) = \frac{2 x}{3 x^{2} + 1}$
$f^{'} (x) = \frac{{(2 x)}^{'} (3 x^{2} + 1) - 2 x {(3 x^{2} + 1)}^{'}}{{(3 x^{2} + 1)}^{2}}$
$= \frac{2 {(x)}^{'} (3 x^{2} + 1) - 2 x ({(3 x^{2})}^{'} + {(1)}^{'})}{{(3 x^{2} + 1)}^{2}}$
$= \frac{2 (3 x^{2} + 1) - 2 x (6 x + 0)}{{(3 x^{2} + 1)}^{2}}$
$= \frac{6 x^{2} + 2 - 12 x^{2}}{{(3 x^{2} + 1)}^{2}}$
$= \frac{2 - 6 x^{2}}{{(3 x^{2} + 1)}^{2}}$
Hence, the derivative of given function is $\frac{2 - 6 x^{2}}{{(3 x^{2} + 1)}^{2}}$ .

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Calculus

Didn't find what you were looking for?