derivative of y= 1/(sqrt(a^2-x^2))

We save the definition:

y = \frac{1}{\sqrt{a^{2} - x^{2}}}

The expression

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

contains several variables: the variable x is chosen for derivation

To simplify differentiation, we rewrite

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

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

Let's perform the differentiation:

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

We differentiate using:

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

(where f =

a^{2} - x^{2}

)

ⓘ The derivative of x^n is n*x^(n-1):

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

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

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

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

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

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

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

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

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

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

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

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

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

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

ⓘ The derivative of a constant is zero:

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

0

■The derivative of x^n is n*x^(n-1):

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

- 2 x

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

ⓘ Adding 0 changes nothing:

0 - 2 x

- 2 x

■Two minus signs cancel out:

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

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

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

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

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

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

▶Derivative:

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