What is (\partial)/(\partial x)(9xsin(5x^2y)) ?

The answer to (\partial)/(\partial x)(9xsin(5x^2y)) is 9(sin(5x^2y)+10x^2ycos(5x^2y))

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(\partial)/(\partial x)(9xsin(5x^2y))

\frac{\partial}{\partial x} (9 x sin (5 x^{2} y))

9 (sin (5 x^{2} y) + 10 x^{2} y cos (5 x^{2} y))

Solution steps

\frac{\partial}{\partial x} (9 x sin (5 x^{2} y))

Treat

y

as a constant

Take the constant out:

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

= 9 \frac{\partial}{\partial x} (x sin (5 x^{2} y))

Apply the Product Rule:

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

f = x, g = sin (5 x^{2} y)

= 9 (\frac{\partial}{\partial x} (x) sin (5 x^{2} y) + \frac{\partial}{\partial x} (sin (5 x^{2} y)) x)

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

\frac{\partial}{\partial x} (sin (5 x^{2} y)) = cos (5 x^{2} y) \cdot 10 y x

= 9 (1 \cdot sin (5 x^{2} y) + cos (5 x^{2} y) \cdot 10 y xx)

Simplify

9 (1 \cdot sin (5 x^{2} y) + cos (5 x^{2} y) \cdot 10 y xx) : 9 (sin (5 x^{2} y) + 10 x^{2} y cos (5 x^{2} y))

= 9 (sin (5 x^{2} y) + 10 x^{2} y cos (5 x^{2} y))

x \to \infty lim (8 - e^{x})

\int \frac{3 ( x - 3 ) ^{2}}{16} d x

\frac{( 2 e ^{x} )}{x ^{2}}

\int (x + 7) d x

\frac{d}{d x} (sin (5 x) + cos (5 x))

What is (\partial)/(\partial x)(9xsin(5x^2y)) ?
The answer to (\partial)/(\partial x)(9xsin(5x^2y)) is 9(sin(5x^2y)+10x^2ycos(5x^2y))