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

Let's differentiate:

\frac{d}{d x} (\sqrt{2 - 3 x^{2}})

We differentiate using:

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

(where f =

2 - 3 x^{2}

)

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

\frac{d}{d x} (\sqrt{2 - 3 x^{2}})

\frac{d}{d x} (2 - 3 x^{2}) \cdot \frac{1}{2 \cdot \sqrt{2 - 3 x^{2}}}

\frac{d}{d x} (2 - 3 x^{2}) \cdot \frac{1}{2 \cdot \sqrt{2 - 3 x^{2}}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (2 - 3 x^{2}) \cdot \frac{1}{2 \cdot \sqrt{2 - 3 x^{2}}}

\frac{\frac{d}{d x} (2 - 3 x^{2}) \cdot 1}{2 \cdot \sqrt{2 - 3 x^{2}}}

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

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

\frac{d}{d x} (2 - 3 x^{2})

0 + \frac{d}{d x} (- 3 x^{2})

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

ⓘ Adding 0 changes nothing:

0 + \frac{d}{d x} (- 3 x^{2})

\frac{d}{d x} (- 3 x^{2})

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

\frac{d}{d x} (- 3 x^{2})

-3 \cdot \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{-3 \cdot 2 x}{2 \cdot \sqrt{2 - 3 x^{2}}}

ⅰ

3 \cdot 2

6

\frac{-6 x}{2 \cdot \sqrt{2 - 3 x^{2}}}

- \frac{6 x}{2 \cdot \sqrt{2 - 3 x^{2}}}

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

- \frac{6 x}{2 \cdot \sqrt{2 - 3 x^{2}}}

- \frac{3 x}{\sqrt{2 - 3 x^{2}}}

▶Derivative:

- \frac{3 x}{\sqrt{2 - 3 x^{2}}}