What is (\partial)/(\partial x)(y^3e^{2x}+5sin(xy)) ?

The answer to (\partial)/(\partial x)(y^3e^{2x}+5sin(xy)) is y^3e^{2x}*2+5cos(xy)y

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(\partial)/(\partial x)(y^3e^{2x}+5sin(xy))

\frac{\partial}{\partial x} (y^{3} e^{2 x} + 5 sin (x y))

y^{3} e^{2 x} \cdot 2 + 5 cos (x y) y

Solution steps

\frac{\partial}{\partial x} (y^{3} e^{2 x} + 5 sin (x y))

Treat

y

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial x} (y^{3} e^{2 x}) + \frac{\partial}{\partial x} (5 sin (x y))

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

\frac{\partial}{\partial x} (5 sin (x y)) = 5 cos (x y) y

= y^{3} e^{2 x} \cdot 2 + 5 cos (x y) y

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

y^{^{''}} + 6 y = 0, y^{^{'}} (1) = 5

\frac{d y}{d x} = \frac{( x ^{2} + 1 )}{y ^{2} + 1}, y (0) = 1

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

x \to 3 lim (\frac{3 x ^{2} - 1 - 2}{x - 3})

What is (\partial)/(\partial x)(y^3e^{2x}+5sin(xy)) ?
The answer to (\partial)/(\partial x)(y^3e^{2x}+5sin(xy)) is y^3e^{2x}*2+5cos(xy)y