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

Let's perform the differentiation:

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

We differentiate using:

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

(where f =

e^{x} + 1

)

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

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

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

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

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

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

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

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

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

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

\frac{\frac{d}{d x} (e^{x}) + 0}{2 \cdot \sqrt{e^{x} + 1}}

ⓘ Adding 0 changes nothing:

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

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

■The derivative of exp(x) is exp(x):

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

e^{x}

▶Derivative:

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