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

The answer to (\partial)/(\partial x)(x^3sin(x^2)) is 3x^2sin(x^2)+2x^4cos(x^2)

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(\partial)/(\partial x)(x^3sin(x^2))

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

3 x^{2} sin (x^{2}) + 2 x^{4} cos (x^{2})

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

cos (x^{2}) \cdot 2 x x^{3} = 2 x^{4} cos (x^{2})

= 3 x^{2} sin (x^{2}) + 2 x^{4} cos (x^{2})

\int_{1}^{2} \frac{9}{x ^{3} + 2 x} d x

y^{^{''}} + 5 y^{^{'}} + 6 y = 0

y^{^{''}} + 3 y = 9

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

\int 4 r^{2} + 1 r d r

What is (\partial)/(\partial x)(x^3sin(x^2)) ?
The answer to (\partial)/(\partial x)(x^3sin(x^2)) is 3x^2sin(x^2)+2x^4cos(x^2)