derivative of x^2sqrt(3x+5) — step-by-step solution

1) Let's differentiate:

\frac{d}{d x} (x^{2} \cdot \sqrt{3 x + 5})

We differentiate

u1

x^{2}

and

v2

\sqrt{3 x + 5}

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} (x^{2})

ⓘ The derivative of x^n is n*x^(n-1):

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

2 x

2 x

3) Let's differentiate

v2

\frac{d}{d x} (\sqrt{3 x + 5})

Apply the differentiation rule:

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

with f =

3 x + 5

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

\frac{d}{d x} (\sqrt{3 x + 5})

\frac{d}{d x} (3 x + 5) \cdot \frac{1}{2 \cdot \sqrt{3 x + 5}}

\frac{d}{d x} (3 x + 5) \cdot \frac{1}{2 \cdot \sqrt{3 x + 5}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (3 x + 5) \cdot \frac{1}{2 \cdot \sqrt{3 x + 5}}

\frac{\frac{d}{d x} (3 x + 5) \cdot 1}{2 \cdot \sqrt{3 x + 5}}

\frac{\frac{d}{d x} (3 x + 5)}{2 \cdot \sqrt{3 x + 5}}

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

\frac{d}{d x} (3 x + 5)

\frac{d}{d x} (3 x) + 0

\frac{\frac{d}{d x} (3 x) + 0}{2 \cdot \sqrt{3 x + 5}}

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (3 x) + 0

\frac{d}{d x} (3 x)

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

\frac{d}{d x} (3 x)

3

\frac{3}{2 \cdot \sqrt{3 x + 5}}

4) Then you can continue the differentiation:

ⅰ

\frac{d}{d x} (x^{2} \cdot \sqrt{3 x + 5})

2 x \cdot \sqrt{3 x + 5} + \frac{x^{2} \cdot 3}{2 \cdot \sqrt{3 x + 5}}

2 x \cdot \sqrt{3 x + 5} + \frac{x^{2} \cdot 3}{2 \cdot \sqrt{3 x + 5}}

▶Derivative:

\frac{3 x^{2}}{2 \cdot \sqrt{3 x + 5}} + 2 x \cdot \sqrt{3 x + 5}