What is the derivative of sqrt((a^2+x^2/(a^2-x^2))) ?

The derivative of sqrt((a^2+x^2/(a^2-x^2))) is (2a^2x)/(sqrt(x^2+a^2)(-x^2+a^2)^{3/2)}

What is the first derivative of sqrt((a^2+x^2/(a^2-x^2))) ?

The first derivative of sqrt((a^2+x^2/(a^2-x^2))) is (2a^2x)/(sqrt(x^2+a^2)(-x^2+a^2)^{3/2)}

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

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

Solution steps

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

Apply the chain rule:

\frac{1}{2 \frac{a ^{2} + x ^{2}}{a ^{2} - x ^{2}}} \frac{d}{d x} (\frac{a ^{2} + x ^{2}}{a ^{2} - x ^{2}})

= \frac{1}{2 \frac{a ^{2} + x ^{2}}{a ^{2} - x ^{2}}} \frac{d}{d x} (\frac{a ^{2} + x ^{2}}{a ^{2} - x ^{2}})

\frac{d}{d x} (\frac{a ^{2} + x ^{2}}{a ^{2} - x ^{2}}) = \frac{4 a ^{2} x}{( a ^{2} - x ^{2} ) ^{2}}

= \frac{1}{2 \frac{a ^{2} + x ^{2}}{a ^{2} - x ^{2}}} \cdot \frac{4 a ^{2} x}{( a ^{2} - x ^{2} ) ^{2}}

Simplify

\frac{1}{2 \frac{a ^{2} + x ^{2}}{a ^{2} - x ^{2}}} \cdot \frac{4 a ^{2} x}{( a ^{2} - x ^{2} ) ^{2}} : \frac{2 a ^{2} x}{x ^{2} + a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{3}{2}}}

= \frac{2 a ^{2} x}{x ^{2} + a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{3}{2}}}

What is the derivative of sqrt((a^2+x^2/(a^2-x^2))) ?
The derivative of sqrt((a^2+x^2/(a^2-x^2))) is (2a^2x)/(sqrt(x^2+a^2)(-x^2+a^2)^{3/2)}
What is the first derivative of sqrt((a^2+x^2/(a^2-x^2))) ?
The first derivative of sqrt((a^2+x^2/(a^2-x^2))) is (2a^2x)/(sqrt(x^2+a^2)(-x^2+a^2)^{3/2)}