What is (\partial)/(\partial y)(sqrt(x^y+ln(z))) ?

The answer to (\partial)/(\partial y)(sqrt(x^y+ln(z))) is (x^yln(x))/(2sqrt(x^y+ln(z)))

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(\partial)/(\partial y)(sqrt(x^y+ln(z)))

\frac{\partial}{\partial y} (x^{y} + ln (z))

\frac{x ^{y} ln ( x )}{2 x ^{y} + ln ( z )}

Solution steps

\frac{\partial}{\partial y} (x^{y} + ln (z))

Treat

x, z

as constants

Apply the chain rule:

\frac{1}{2 x ^{y} + ln ( z )} \frac{\partial}{\partial y} (x^{y} + ln (z))

= \frac{1}{2 x ^{y} + ln ( z )} \frac{\partial}{\partial y} (x^{y} + ln (z))

\frac{\partial}{\partial y} (x^{y} + ln (z)) = x^{y} ln (x)

= \frac{1}{2 x ^{y} + ln ( z )} x^{y} ln (x)

Simplify

\frac{1}{2 x ^{y} + ln ( z )} x^{y} ln (x) : \frac{x ^{y} ln ( x )}{2 x ^{y} + ln ( z )}

= \frac{x ^{y} ln ( x )}{2 x ^{y} + ln ( z )}

\frac{d y}{d t} - 5 y = sin (2 t)

y = 47 (2 t^{3} - 9)^{\frac{6}{5}}

\int \frac{3}{5 y - 4} d y

x \to 0 lim (\frac{6 x n - 5 x n - 1 - 7 x n - 2}{- 6 x n - 4 x n - 2})

f (x) = (x^{4} + 3 x^{2} - 2)^{5}

What is (\partial)/(\partial y)(sqrt(x^y+ln(z))) ?
The answer to (\partial)/(\partial y)(sqrt(x^y+ln(z))) is (x^yln(x))/(2sqrt(x^y+ln(z)))