derivative of sqrt(x)(2x+4) — step-by-step solution

⚠ sqrt(x)(2x+4) is read as sqrt(x)*(2x+4): use (...)*(...) for multiplication, use (...)^(...) for a power

1) Let's perform the differentiation:

\frac{d}{d x} (\sqrt{x} \cdot (2 x + 4))

We differentiate

u1

\sqrt{x}

and

v2

2 x + 4

using:

\frac{d}{d x} (u1 \cdot v2) = \frac{d}{d x} (u1) \cdot v2 + u1 \cdot \frac{d}{d x} (v2)

2) Let's differentiate

u1

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

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

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

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

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

3) Let's perform the differentiation

v2

\frac{d}{d x} (2 x + 4)

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

\frac{d}{d x} (2 x + 4)

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

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

ⓘ Adding 0 changes nothing:

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

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

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

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

2

2

4) We evaluate the resulting derivative (

u1 \cdot v2

)':

\frac{1}{2 \sqrt{x}} \cdot (2 x + 4) + \sqrt{x} \cdot 2

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{1}{2 \sqrt{x}} \cdot (2 x + 4)

\frac{1 \cdot (2 x + 4)}{2 \sqrt{x}}

\frac{2 x + 4}{2 \sqrt{x}}

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

\frac{2 x + 4}{2 \sqrt{x}}

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

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

5) Then you can continue the derivation:

ⅰ

\frac{d}{d x} (\sqrt{x} \cdot (2 x + 4))

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

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

▶Derivative:

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