derivative of a/(bx^2) — step-by-step solution

⚠ bx is read as the product b*x: for the variable bx, write bx_

The expression

\frac{a}{b x^{2}}

contains several variables: the variable x is chosen for differentiation

1) Let's perform the differentiation:

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

For

p1

a

and

q2

b 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 perform the differentiation

p1

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

ⓘ The derivative of a constant is zero:

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

0

0

3) Let's differentiate

q2

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

ⓘ For differentiation with respect to

x

b

is treated as constant. A constant factor stays in front of the derivative, (k*u)' = k*u':

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

b \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

b \cdot 2 x

4) Let's evaluate the resulting derivative (

\frac{p1}{q2}

)':

\frac{0 \cdot b x^{2} - a \cdot b \cdot 2 x}{{(b x^{2})}^{2}}

ⓘ A product with a zero factor is zero:

0 \cdot b x^{2}

0

\frac{0 - a b \cdot 2 x}{{(b x^{2})}^{2}}

ⓘ Adding 0 changes nothing:

0 - a b \cdot 2 x

- a b \cdot 2 x

- \frac{a b \cdot 2 x}{{(b x^{2})}^{2}}

ⓘ The power of a product is the product of the powers:

{(b x^{2})}^{2}

b^{2} \cdot {(x^{2})}^{2}

- \frac{a b \cdot 2 x}{b^{2} \cdot {(x^{2})}^{2}}

We simplify the fraction:

\frac{b}{b^{2}}

by the common factor:

b

to get the new fraction:

\frac{1}{b}

\frac{2 \cdot (- a x)}{b \cdot {(x^{2})}^{2}}

ⓘ The minus sign is placed in front of the product:

2 \cdot (- a)

-2 a

■A power of a power multiplies the exponents, (a^n)^m = a^(n*m):

{(x^{2})}^{2}

x^{2 \cdot 2}

- \frac{2 a x}{b x^{2 \cdot 2}}

ⅰ

2 \cdot 2

4

- \frac{2 a x}{b x^{4}}

We divide the numerator and denominator of

\frac{x}{x^{4}}

by the common factor:

x

, which gives the new fraction:

\frac{1}{x^{3}}

\frac{-2 a}{b x^{3}}

- \frac{2 a}{b x^{3}}

5) Then you can continue the differentiation:

ⅰ

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

- \frac{2 a}{b x^{3}}

▶Derivative:

- \frac{2 a}{b x^{3}}