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

The answer to (\partial)/(\partial x)(2xy^3cos(x^2y^3)) is 2y^3(cos(x^2y^3)-2x^2y^3sin(x^2y^3))

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(\partial)/(\partial x)(2xy^3cos(x^2y^3))

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

2 y^{3} (cos (x^{2} y^{3}) - 2 x^{2} y^{3} sin (x^{2} y^{3}))

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

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

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

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

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (cos (x^{2} y^{3})) = - sin (x^{2} y^{3}) \cdot 2 y^{3} x

= 2 y^{3} (1 \cdot cos (x^{2} y^{3}) + (- sin (x^{2} y^{3}) \cdot 2 y^{3} x) x)

Simplify

2 y^{3} (1 \cdot cos (x^{2} y^{3}) + (- sin (x^{2} y^{3}) \cdot 2 y^{3} x) x) : 2 y^{3} (cos (x^{2} y^{3}) - 2 x^{2} y^{3} sin (x^{2} y^{3}))

= 2 y^{3} (cos (x^{2} y^{3}) - 2 x^{2} y^{3} sin (x^{2} y^{3}))

\frac{\partial}{\partial y} (3 tan (x y))

\frac{d y}{d x} = 3 (y^{2} + 1)

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

f (x) = tan (x), (\frac{π}{4}, 1)

\int \frac{4}{x ^{2} + 2 x + 5} d x

What is (\partial)/(\partial x)(2xy^3cos(x^2y^3)) ?
The answer to (\partial)/(\partial x)(2xy^3cos(x^2y^3)) is 2y^3(cos(x^2y^3)-2x^2y^3sin(x^2y^3))