What is (\partial)/(\partial x)(5xsin(3x^2y)) ?

The answer to (\partial)/(\partial x)(5xsin(3x^2y)) is 5(sin(3x^2y)+6x^2ycos(3x^2y))

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(\partial)/(\partial x)(5xsin(3x^2y))

\frac{\partial}{\partial x} (5 x sin (3 x^{2} y))

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

Solution steps

\frac{\partial}{\partial x} (5 x sin (3 x^{2} y))

Treat

y

as a constant

Take the constant out:

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

= 5 \frac{\partial}{\partial x} (x sin (3 x^{2} y))

Apply the Product Rule:

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

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

= 5 (\frac{\partial}{\partial x} (x) sin (3 x^{2} y) + \frac{\partial}{\partial x} (sin (3 x^{2} y)) x)

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

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

= 5 (1 \cdot sin (3 x^{2} y) + cos (3 x^{2} y) \cdot 6 y xx)

Simplify

5 (1 \cdot sin (3 x^{2} y) + cos (3 x^{2} y) \cdot 6 y xx) : 5 (sin (3 x^{2} y) + 6 x^{2} y cos (3 x^{2} y))

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

\frac{d}{d x} (sin^{l n (x)} (x))

x \to 2 lim (5 x + 1)

\frac{\partial}{\partial y} ((x^{2} + y^{2})^{\frac{4}{5}})

f (x) = x^{2} - x + 13, at x = 4

f (x) = - \frac{4}{3} x^{\frac{2}{3}}

What is (\partial)/(\partial x)(5xsin(3x^2y)) ?
The answer to (\partial)/(\partial x)(5xsin(3x^2y)) is 5(sin(3x^2y)+6x^2ycos(3x^2y))