What is (\partial)/(\partial x)(3xsin(9x^2y)) ?

The answer to (\partial)/(\partial x)(3xsin(9x^2y)) is 3(sin(9x^2y)+18x^2ycos(9x^2y))

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(\partial)/(\partial x)(3xsin(9x^2y))

\frac{\partial}{\partial x} (3 x sin (9 x^{2} y))

3 (sin (9 x^{2} y) + 18 x^{2} y cos (9 x^{2} y))

Solution steps

\frac{\partial}{\partial x} (3 x sin (9 x^{2} y))

Treat

y

as a constant

Take the constant out:

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

= 3 \frac{\partial}{\partial x} (x sin (9 x^{2} y))

Apply the Product Rule:

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

f = x, g = sin (9 x^{2} y)

= 3 (\frac{\partial}{\partial x} (x) sin (9 x^{2} y) + \frac{\partial}{\partial x} (sin (9 x^{2} y)) x)

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (sin (9 x^{2} y)) = cos (9 x^{2} y) \cdot 18 y x

= 3 (1 \cdot sin (9 x^{2} y) + cos (9 x^{2} y) \cdot 18 y xx)

Simplify

3 (1 \cdot sin (9 x^{2} y) + cos (9 x^{2} y) \cdot 18 y xx) : 3 (sin (9 x^{2} y) + 18 x^{2} y cos (9 x^{2} y))

= 3 (sin (9 x^{2} y) + 18 x^{2} y cos (9 x^{2} y))

x \to \infty lim ((1 + x)^{\frac{3}{x}})

\int \frac{1}{x ^{2} + 36} d x

x \to - \infty lim (\frac{x ^{2} + 2}{x ^{3} + x ^{2} - 1})

x \to 0 lim (7 x^{3} + \frac{1}{4} x^{2})

\frac{\partial}{\partial y} (x \cdot ln (x + 2 y) + \frac{x}{y})

What is (\partial)/(\partial x)(3xsin(9x^2y)) ?
The answer to (\partial)/(\partial x)(3xsin(9x^2y)) is 3(sin(9x^2y)+18x^2ycos(9x^2y))