What is (\partial)/(\partial x)((x^{p(x,y)}+y^{p(x,y)})^{1/(p(x,y))}) ?

The answer to (\partial)/(\partial x)((x^{p(x,y)}+y^{p(x,y)})^{1/(p(x,y))}) is

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(\partial)/(\partial x)((x^{p(x,y)}+y^{p(x,y)})^{1/(p(x,y))})

\frac{\partial}{\partial x} ((x^{p (x, y)} + y^{p (x, y)})^{\frac{1}{p ( x , y )}})

e^{\frac{1}{p ( x , y )} l n (x^{p (x, y)} + y^{p (x, y)})} \frac{1}{p} \cdot \frac{1}{, y} ((- \frac{1}{x ^{2}}) ln (x^{p (x, y)} + y^{p (x, y)}) + \frac{1}{x ^{p (x, y)} + y ^{p (x, y)}} (e^{p (x, y) l n (x)} p ((, y \cdot 1) ln (x) + \frac{1}{x} (x, y)) + e^{p (x, y) l n (y)} p ln (y) (, y \cdot 1)) \frac{1}{x})

Solution steps

\frac{\partial}{\partial x} ((x^{p (x, y)} + y^{p (x, y)})^{\frac{1}{p ( x , y )}})

Treat

p, y

as a constant

Apply exponent rule:

a^{b} = e^{b l n (a)}

(x^{p (x, y)} + y^{p (x, y)})^{\frac{1}{p ( x , y )}} = e^{\frac{1}{p ( x , y )} l n (x^{p (x, y)} + y^{p (x, y)})}

= \frac{\partial}{\partial x} (e^{\frac{1}{p ( x , y )} l n (x^{p (x, y)} + y^{p (x, y)})})

Apply the chain rule:

e^{\frac{1}{p ( x , y )} l n (x^{p (x, y)} + y^{p (x, y)})} \frac{\partial}{\partial x} (\frac{1}{p ( x , y )} ln (x^{p (x, y)} + y^{p (x, y)}))

= e^{\frac{1}{p ( x , y )} l n (x^{p (x, y)} + y^{p (x, y)})} \frac{\partial}{\partial x} (\frac{1}{p ( x , y )} ln (x^{p (x, y)} + y^{p (x, y)}))

\frac{\partial}{\partial x} (\frac{1}{p ( x , y )} ln (x^{p (x, y)} + y^{p (x, y)})) = \frac{1}{p} \cdot \frac{1}{, y} ((- \frac{1}{x ^{2}}) ln (x^{p (x, y)} + y^{p (x, y)}) + \frac{1}{x ^{p (x, y)} + y ^{p (x, y)}} (e^{p (x, y) l n (x)} p ((, y \cdot 1) ln (x) + \frac{1}{x} (x, y)) + e^{p (x, y) l n (y)} p ln (y) (, y \cdot 1)) \frac{1}{x})

= e^{\frac{1}{p ( x , y )} l n (x^{p (x, y)} + y^{p (x, y)})} \frac{1}{p} \cdot \frac{1}{, y} ((- \frac{1}{x ^{2}}) ln (x^{p (x, y)} + y^{p (x, y)}) + \frac{1}{x ^{p (x, y)} + y ^{p (x, y)}} (e^{p (x, y) l n (x)} p ((, y \cdot 1) ln (x) + \frac{1}{x} (x, y)) + e^{p (x, y) l n (y)} p ln (y) (, y \cdot 1)) \frac{1}{x})

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f (x) = x^{2} + 55, at x = 3

What is (\partial)/(\partial x)((x^{p(x,y)}+y^{p(x,y)})^{1/(p(x,y))}) ?
The answer to (\partial)/(\partial x)((x^{p(x,y)}+y^{p(x,y)})^{1/(p(x,y))}) is