derivative of 1/((1-2x^{3/2)}) — step-by-step solution

⚠ 1/((1-2x^{3/2)}) is read as 1/((1-2x^(3/2))): use ^(...) to group an exponent, use * to multiply

To ease differentiation, the expression

\frac{1}{1 - 2 x^{\frac{3}{2}}}

is turned into

{(1 - 2 x^{\frac{3}{2}})}^{-1}

Let's perform the differentiation:

\frac{d}{d x} ({(1 - 2 x^{\frac{3}{2}})}^{-1})

We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

(where f =

1 - 2 x^{\frac{3}{2}}

)

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({(1 - 2 x^{\frac{3}{2}})}^{-1})

-1 \cdot \frac{d}{d x} (1 - 2 x^{\frac{3}{2}}) \cdot {(1 - 2 x^{\frac{3}{2}})}^{-2}

-1 \cdot \frac{d}{d x} (1 - 2 x^{\frac{3}{2}}) \cdot {(1 - 2 x^{\frac{3}{2}})}^{-2}

ⓘ Multiplying by -1 gives the opposite:

-1 \cdot \frac{d}{d x} (1 - 2 x^{\frac{3}{2}}) \cdot {(1 - 2 x^{\frac{3}{2}})}^{-2}

- \frac{d}{d x} (1 - 2 x^{\frac{3}{2}}) \cdot {(1 - 2 x^{\frac{3}{2}})}^{-2}

- \frac{d}{d x} (1 - 2 x^{\frac{3}{2}}) \cdot {(1 - 2 x^{\frac{3}{2}})}^{-2}

ⓘ A negative exponent moves the power to the other side of the fraction bar, for a nonzero base:

- \frac{d}{d x} (1 - 2 x^{\frac{3}{2}}) \cdot {(1 - 2 x^{\frac{3}{2}})}^{-2}

- \frac{\frac{d}{d x} (1 - 2 x^{\frac{3}{2}})}{{(1 - 2 x^{\frac{3}{2}})}^{2}}

- \frac{\frac{d}{d x} (1 - 2 x^{\frac{3}{2}})}{{(1 - 2 x^{\frac{3}{2}})}^{2}}

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

\frac{d}{d x} (1 - 2 x^{\frac{3}{2}})

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

- \frac{0 + \frac{d}{d x} (- 2 x^{\frac{3}{2}})}{{(1 - 2 x^{\frac{3}{2}})}^{2}}

ⓘ Adding 0 changes nothing:

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

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

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

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

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

■The derivative of x^n is n*x^(n-1):

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

\frac{3}{2} \cdot x^{\frac{3}{2} - 1}

- (\frac{-2 \cdot \frac{3}{2} \cdot x^{\frac{3}{2} - 1}}{{(1 - 2 x^{\frac{3}{2}})}^{2}})

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{3}{2} - 1

\frac{3 - 1 \cdot 2}{2}

ⅰ

2 \cdot \frac{3}{2}

1 \cdot 3

- (\frac{-1 \cdot 3 \cdot x^{\frac{3 - 1 \cdot 2}{2}}}{{(1 - 2 x^{\frac{3}{2}})}^{2}})

ⓘ Multiplications are done before additions and subtractions:

1 \cdot 2

2

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

-1 \cdot 3 x^{\frac{3 - 2}{2}}

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

- (\frac{- 3 x^{\frac{3 - 2}{2}}}{{(1 - 2 x^{\frac{3}{2}})}^{2}})

ⓘ Two minus signs cancel out:

- (\frac{- 3 x^{\frac{3 - 2}{2}}}{{(1 - 2 x^{\frac{3}{2}})}^{2}})

\frac{3 x^{\frac{3 - 2}{2}}}{{(1 - 2 x^{\frac{3}{2}})}^{2}}

\frac{3 x^{\frac{3 - 2}{2}}}{{(1 - 2 x^{\frac{3}{2}})}^{2}}

ⓘ The numerator and the denominator are computed before the division:

3 - 2

1

▶Derivative:

\frac{3 x^{\frac{1}{2}}}{{(1 - 2 x^{\frac{3}{2}})}^{2}}