What is (\partial)/(\partial x)(48x^3cos(6x^2y)) ?

The answer to (\partial)/(\partial x)(48x^3cos(6x^2y)) is 48(3x^2cos(6x^2y)-12x^4ysin(6x^2y))

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(\partial)/(\partial x)(48x^3cos(6x^2y))

\frac{\partial}{\partial x} (48 x^{3} cos (6 x^{2} y))

48 (3 x^{2} cos (6 x^{2} y) - 12 x^{4} y sin (6 x^{2} y))

Solution steps

\frac{\partial}{\partial x} (48 x^{3} cos (6 x^{2} y))

Treat

y

as a constant

Take the constant out:

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

= 48 \frac{\partial}{\partial x} (x^{3} cos (6 x^{2} y))

Apply the Product Rule:

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

f = x^{3}, g = cos (6 x^{2} y)

= 48 (\frac{\partial}{\partial x} (x^{3}) cos (6 x^{2} y) + \frac{\partial}{\partial x} (cos (6 x^{2} y)) x^{3})

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

\frac{\partial}{\partial x} (cos (6 x^{2} y)) = - sin (6 x^{2} y) \cdot 12 y x

= 48 (3 x^{2} cos (6 x^{2} y) + (- sin (6 x^{2} y) \cdot 12 y x) x^{3})

Simplify

48 (3 x^{2} cos (6 x^{2} y) + (- sin (6 x^{2} y) \cdot 12 y x) x^{3}) : 48 (3 x^{2} cos (6 x^{2} y) - 12 x^{4} y sin (6 x^{2} y))

= 48 (3 x^{2} cos (6 x^{2} y) - 12 x^{4} y sin (6 x^{2} y))

What is (\partial)/(\partial x)(48x^3cos(6x^2y)) ?
The answer to (\partial)/(\partial x)(48x^3cos(6x^2y)) is 48(3x^2cos(6x^2y)-12x^4ysin(6x^2y))