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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

= 6 \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)

= 6 (\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

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

Simplify

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

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

y = x^{2}, y = 3

\int_{0}^{\frac{π}{4}} sec^{2} (x) d x

\frac{d}{d s} (ln (s + \frac{s ^{2} + 4}{2}) 2)

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

4 x + 20

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