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

To ease differentiation, the expression

\frac{2}{\sqrt{x^{2} + 1}}

is turned into

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

Let's perform the differentiation:

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

x^{2} + 1

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

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

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

■The derivative of x^n is n*x^(n-1):

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

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

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

ⓘ Multiplying by -1 gives the opposite:

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

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

ⅰ

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

-1

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

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

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

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

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

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

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

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

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

ⓘ Adding 0 changes nothing:

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

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

■The derivative of x^n is n*x^(n-1):

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

2 x

▶Derivative:

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