derivative of (2x/(3+x^2)) — step-by-step solution

1) Let's perform the differentiation:

\frac{d}{d x} (\frac{2 x}{3 + x^{2}})

For

p1

2 x

and

q2

3 + x^{2}

, use the differentiation rule:

\frac{d}{d x} (\frac{p1}{q2}) = \frac{\frac{d}{d x} (p1) \cdot q2 - p1 \cdot \frac{d}{d x} (q2)}{{q2}^{2}}

2) Let's differentiate

p1

\frac{d}{d x} (2 x)

ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (2 x)

2

2

3) Let's perform the differentiation

q2

\frac{d}{d x} (3 + x^{2})

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (3 + x^{2})

0 + \frac{d}{d x} (x^{2})

0 + \frac{d}{d x} (x^{2})

ⓘ Adding 0 changes nothing:

0 + \frac{d}{d x} (x^{2})

\frac{d}{d x} (x^{2})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{2})

2 x

2 x

4) We evaluate the resulting derivative (

\frac{p1}{q2}

)':

\frac{2 \cdot (3 + x^{2}) - 2 x \cdot 2 x}{{(3 + x^{2})}^{2}}

ⓘ Multiplications are done before additions and subtractions:

2 \cdot 2

4

■Multiplying powers with the same base adds the exponents:

x \cdot x

x^{2}

\frac{2 \cdot (3 + x^{2}) - 4 x^{2}}{{(3 + x^{2})}^{2}}

ⓘ Expand by multiplying each term inside the parentheses:

2 \cdot (3 + x^{2})

6 + 2 x^{2}

\frac{6 + 2 x^{2} - 4 x^{2}}{{(3 + x^{2})}^{2}}

ⓘ Like terms are grouped by adding their coefficients:

2 x^{2} - 4 x^{2}

-2 x^{2}

\frac{6 - 2 x^{2}}{{(3 + x^{2})}^{2}}

5) Then you can resume the differentiation:

ⅰ

\frac{d}{d x} (\frac{2 x}{3 + x^{2}})

\frac{6 - 2 x^{2}}{{(3 + x^{2})}^{2}}

\frac{6 - 2 x^{2}}{{(3 + x^{2})}^{2}}

▶Derivative:

\frac{6 - 2 x^{2}}{{(x^{2} + 3)}^{2}}