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

The expression

\sqrt{y^{2} - x^{2}}

contains several variables: the variable x is chosen for differentiation

Let's perform the differentiation:

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

We differentiate using:

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

(where f =

y^{2} - x^{2}

)

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

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

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

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

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

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

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

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

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

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

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

ⓘ The derivative of a constant is zero:

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

0

■The derivative of x^n is n*x^(n-1):

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

- 2 x

\frac{0 - 2 x}{2 \cdot \sqrt{y^{2} - x^{2}}}

ⓘ Adding 0 changes nothing:

0 - 2 x

- 2 x

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

- \frac{2 x}{2 \cdot \sqrt{y^{2} - x^{2}}}

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

▶Derivative:

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