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

The answer to (\partial)/(\partial x)(2x^3y^2cos(6x^3y)) is 2y^2(3x^2cos(6x^3y)-18x^5ysin(6x^3y))

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

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

2 y^{2} (3 x^{2} cos (6 x^{3} y) - 18 x^{5} y sin (6 x^{3} y))

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

\frac{\partial}{\partial x} (cos (6 x^{3} y)) = - sin (6 x^{3} y) \cdot 18 y x^{2}

= 2 y^{2} (3 x^{2} cos (6 x^{3} y) + (- sin (6 x^{3} y) \cdot 18 y x^{2}) x^{3})

Simplify

2 y^{2} (3 x^{2} cos (6 x^{3} y) + (- sin (6 x^{3} y) \cdot 18 y x^{2}) x^{3}) : 2 y^{2} (3 x^{2} cos (6 x^{3} y) - 18 x^{5} y sin (6 x^{3} y))

= 2 y^{2} (3 x^{2} cos (6 x^{3} y) - 18 x^{5} y sin (6 x^{3} y))

\int \frac{7}{sec ( x ) - 1} d x

\int \frac{e ^{2 x}}{e ^{5 - x}} d x

x \to \infty lim (67)

\int_{- 2}^{x} \frac{x + 2}{3} d x

d er i v a t i v e \frac{3}{x ^{9}}

What is (\partial)/(\partial x)(2x^3y^2cos(6x^3y)) ?
The answer to (\partial)/(\partial x)(2x^3y^2cos(6x^3y)) is 2y^2(3x^2cos(6x^3y)-18x^5ysin(6x^3y))