What is the derivative of x^3sin(x^2) ?

The derivative of x^3sin(x^2) is 3x^2sin(x^2)+2x^4cos(x^2)

What is the first derivative of x^3sin(x^2) ?

The first derivative of x^3sin(x^2) is 3x^2sin(x^2)+2x^4cos(x^2)

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

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

Solution steps

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

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

\frac{d}{d 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})

What is the derivative of x^3sin(x^2) ?
The derivative of x^3sin(x^2) is 3x^2sin(x^2)+2x^4cos(x^2)
What is the first derivative of x^3sin(x^2) ?
The first derivative of x^3sin(x^2) is 3x^2sin(x^2)+2x^4cos(x^2)