What is (\partial)/(\partial x)(x^4sin(xy^3)) ?

The answer to (\partial)/(\partial x)(x^4sin(xy^3)) is 4x^3sin(xy^3)+cos(xy^3)y^3x^4

(\partial)/(\partial x)(x^4sin(xy^3))

\frac{\partial}{\partial x} (x^{4} sin (x y^{3}))

4 x^{3} sin (x y^{3}) + cos (x y^{3}) y^{3} x^{4}

Solution steps

\frac{\partial}{\partial x} (x^{4} sin (x y^{3}))

Treat

y

as a constant

Apply the Product Rule:

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

f = x^{4}, g = sin (x y^{3})

= \frac{\partial}{\partial x} (x^{4}) sin (x y^{3}) + \frac{\partial}{\partial x} (sin (x y^{3})) x^{4}

\frac{\partial}{\partial x} (x^{4}) = 4 x^{3}

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

= 4 x^{3} sin (x y^{3}) + cos (x y^{3}) y^{3} x^{4}

\int \frac{( arctan ( 5 x ) )}{1 + 25 x ^{2}} d x

\frac{d}{d x} (- x^{5})

\int (1 + x^{3})^{5} x^{2} d x

\frac{d}{d y} (x y)

x^{2} y^{^{'}} + 4 x y = \frac{sin ( 3 x )}{x ^{2}}

What is (\partial)/(\partial x)(x^4sin(xy^3)) ?
The answer to (\partial)/(\partial x)(x^4sin(xy^3)) is 4x^3sin(xy^3)+cos(xy^3)y^3x^4