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

1) Let's perform the differentiation:

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

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

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

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

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

We differentiate

p1

x^{2}

and

q2

3 x^{2} + 2

using:

\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} (x^{2})

ⓘ The derivative of x^n is n*x^(n-1):

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

2 x

2 x

3) Let's differentiate

q2

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

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

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

\frac{d}{d x} (3 x^{2}) + 0

\frac{d}{d x} (3 x^{2}) + 0

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (3 x^{2}) + 0

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

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

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

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

■The derivative of x^n is n*x^(n-1):

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

2 x

3 \cdot 2 x

ⅰ

3 \cdot 2

6

6 x

4) We evaluate the resulting derivative (

\frac{p1}{q2}

)':

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

ⓘ Multiplying powers with the same base adds the exponents:

x^{2} x

x^{3}

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

ⓘ The minus sign of the fraction goes into the numerator or the denominator, changing the sign of each of its terms:

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

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

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

ⓘ What the terms have in common is factored out, so that the fraction can be simplified:

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

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

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

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (3 x^{2} + 2)

- 3 x^{2} - 2

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

ⓘ Like terms are grouped by adding their coefficients:

- 3 x^{2} + 3 x^{2}

0

\frac{2 x \cdot (0 - 2)}{{(3 x^{2} + 2)}^{2}}

ⓘ Adding 0 changes nothing:

0 - 2

-2

■A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

2 \cdot (-2)

-4

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

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

5) You can then resume the differentiation:

ⅰ

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

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

▶Derivative:

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