What is the d/(dz)(ln(sqrt((c^2-z^2)/(c^2+z^2)))) ?

The d/(dz)(ln(sqrt((c^2-z^2)/(c^2+z^2)))) is -(2c^2z)/((z^2+c^2)(-z^2+c^2))

What is the first d/(dz)(ln(sqrt((c^2-z^2)/(c^2+z^2)))) ?

The first d/(dz)(ln(sqrt((c^2-z^2)/(c^2+z^2)))) is -(2c^2z)/((z^2+c^2)(-z^2+c^2))

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\frac{d}{d z} (ln (\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}))

- \frac{2 c ^{2} z}{( z ^{2} + c ^{2} ) ( - z ^{2} + c ^{2} )}

Solution steps

\frac{d}{d z} (ln (\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}))

Apply the chain rule:

\frac{1}{\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}} \frac{d}{d z} (\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}})

= \frac{1}{\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}} \frac{d}{d z} (\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}})

\frac{d}{d z} (\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}) = - \frac{2 c ^{2} z}{- z ^{2} + c ^{2} ( z ^{2} + c ^{2} ) ^{\frac{3}{2}}}

= \frac{1}{\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}} (- \frac{2 c ^{2} z}{- z ^{2} + c ^{2} ( z ^{2} + c ^{2} ) ^{\frac{3}{2}}})

Simplify

\frac{1}{\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}} (- \frac{2 c ^{2} z}{- z ^{2} + c ^{2} ( z ^{2} + c ^{2} ) ^{\frac{3}{2}}}) : - \frac{2 c ^{2} z}{( z ^{2} + c ^{2} ) ( - z ^{2} + c ^{2} )}

= - \frac{2 c ^{2} z}{( z ^{2} + c ^{2} ) ( - z ^{2} + c ^{2} )}

y^{^{''}} + 4 y^{^{'}} + 6 y = 1 + e^{- t}, y (0) = 0, y^{^{'}} (0) = 0

\frac{\partial}{\partial x} ((4 x + 7 y)^{10})

\int x^{2} + x + 2 d x

\int \frac{13 t ^{7} + t ^{5}}{t ^{\frac{3}{2}}} d t

x \to 0 lim (x^{x^{2}})

What is the d/(dz)(ln(sqrt((c^2-z^2)/(c^2+z^2)))) ?
The d/(dz)(ln(sqrt((c^2-z^2)/(c^2+z^2)))) is -(2c^2z)/((z^2+c^2)(-z^2+c^2))
What is the first d/(dz)(ln(sqrt((c^2-z^2)/(c^2+z^2)))) ?
The first d/(dz)(ln(sqrt((c^2-z^2)/(c^2+z^2)))) is -(2c^2z)/((z^2+c^2)(-z^2+c^2))