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

To ease differentiation, the expression

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

is turned into

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

Let's differentiate:

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

Apply the differentiation rule:

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

with f =

1 - x^{2}

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

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

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

■The derivative of x^n is n*x^(n-1):

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

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

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

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

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

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

ⅰ

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

- \frac{3}{2}

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

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

ⅰ

\frac{3}{2} \cdot (-2)

3 \cdot (-1)

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

ⓘ Two minus signs cancel out, the product of two negative numbers is positive:

-3 \cdot (-1)

3

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

ⓘ The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

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

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

▶Derivative:

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