What is (\partial)/(\partial x)(4xcos(5xy)) ?

The answer to (\partial)/(\partial x)(4xcos(5xy)) is 4(cos(5xy)-5xysin(5xy))

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(\partial)/(\partial x)(4xcos(5xy))

\frac{\partial}{\partial x} (4 x cos (5 x y))

4 (cos (5 x y) - 5 x y sin (5 x y))

Solution steps

\frac{\partial}{\partial x} (4 x cos (5 x y))

Treat

y

as a constant

Take the constant out:

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

= 4 \frac{\partial}{\partial x} (x cos (5 x y))

Apply the Product Rule:

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

f = x, g = cos (5 x y)

= 4 (\frac{\partial}{\partial x} (x) cos (5 x y) + \frac{\partial}{\partial x} (cos (5 x y)) x)

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

\frac{\partial}{\partial x} (cos (5 x y)) = - sin (5 x y) \cdot 5 y

= 4 (1 \cdot cos (5 x y) + (- sin (5 x y) \cdot 5 y) x)

Simplify

= 4 (cos (5 x y) - 5 x y sin (5 x y))

What is (\partial)/(\partial x)(4xcos(5xy)) ?
The answer to (\partial)/(\partial x)(4xcos(5xy)) is 4(cos(5xy)-5xysin(5xy))