What is (\partial)/(\partial x)(x^3sec^2(xy)) ?

The answer to (\partial)/(\partial x)(x^3sec^2(xy)) is 3x^2sec^2(xy)+2ysec^2(yx)tan(yx)x^3

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(\partial)/(\partial x)(x^3sec^2(xy))

\frac{\partial}{\partial x} (x^{3} sec^{2} (x y))

3 x^{2} sec^{2} (x y) + 2 y sec^{2} (y x) tan (y x) x^{3}

Solution steps

\frac{\partial}{\partial x} (x^{3} sec^{2} (x y))

Treat

y

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{3}, g = sec^{2} (x y)

= \frac{\partial}{\partial x} (x^{3}) sec^{2} (x y) + \frac{\partial}{\partial x} (sec^{2} (x y)) x^{3}

\frac{\partial}{\partial x} (x^{3}) = 3 x^{2}

\frac{\partial}{\partial x} (sec^{2} (x y)) = 2 y sec^{2} (y x) tan (y x)

= 3 x^{2} sec^{2} (x y) + 2 y sec^{2} (y x) tan (y x) x^{3}

What is (\partial)/(\partial x)(x^3sec^2(xy)) ?
The answer to (\partial)/(\partial x)(x^3sec^2(xy)) is 3x^2sec^2(xy)+2ysec^2(yx)tan(yx)x^3