differentiate sqrt(x^2+1) — step-by-step solution

Let's differentiate:

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

Apply the differentiation rule:

\frac{d}{d x} (\sqrt{f}) = \frac{d}{d x} (f) \cdot \frac{1}{2 \sqrt{f}}

with f =

x^{2} + 1

ⓘ The derivative of sqrt(x) is 1/(2*sqrt(x)):

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

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

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

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

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

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

■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}{2 \cdot \sqrt{x^{2} + 1}}

ⓘ 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

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

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

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

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

▶Derivative:

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