derivative of f(x)=sqrt(x+8) — step-by-step solution

We save the definition:

f (x) = \sqrt{x + 8}

Let's perform the differentiation:

\frac{d}{d x} (\sqrt{x + 8})

We differentiate using:

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

(where f =

x + 8

)

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

\frac{d}{d x} (\sqrt{x + 8})

\frac{d}{d x} (x + 8) \cdot \frac{1}{2 \cdot \sqrt{x + 8}}

\frac{d}{d x} (x + 8) \cdot \frac{1}{2 \cdot \sqrt{x + 8}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (x + 8) \cdot \frac{1}{2 \cdot \sqrt{x + 8}}

\frac{\frac{d}{d x} (x + 8) \cdot 1}{2 \cdot \sqrt{x + 8}}

\frac{\frac{d}{d x} (x + 8)}{2 \cdot \sqrt{x + 8}}

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

\frac{d}{d x} (x + 8)

1 + 0

\frac{1 + 0}{2 \cdot \sqrt{x + 8}}

ⓘ Adding 0 changes nothing:

1 + 0

1

▶Derivative:

\frac{1}{2 \cdot \sqrt{x + 8}}