What is (\partial)/(\partial x)(e^{xy}cos(bx+c)) ?

The answer to (\partial)/(\partial x)(e^{xy}cos(bx+c)) is e^{xy}ycos(bx+c)-be^{xy}sin(bx+c)

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(\partial)/(\partial x)(e^{xy}cos(bx+c))

\frac{\partial}{\partial x} (e^{x y} cos (b x + c))

e^{x y} y cos (b x + c) - b e^{x y} sin (b x + c)

Solution steps

\frac{\partial}{\partial x} (e^{x y} cos (b x + c))

Treat

b, c, y

as a constant

Apply the Product Rule:

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

f = e^{x y}, g = cos (b x + c)

= \frac{\partial}{\partial x} (e^{x y}) cos (b x + c) + \frac{\partial}{\partial x} (cos (b x + c)) e^{x y}

\frac{\partial}{\partial x} (e^{x y}) = e^{x y} y

\frac{\partial}{\partial x} (cos (b x + c)) = - sin (b x + c) b

= e^{x y} y cos (b x + c) + (- sin (b x + c) b) e^{x y}

Simplify

= e^{x y} y cos (b x + c) - b e^{x y} sin (b x + c)

\frac{d y}{d x} = 2 + y - 2 x + 4

\frac{d y}{d x}, 2 x y = 6

(8.3) (3.5)

\int (x^{2} + 4)^{5} 3 x d x

\frac{\partial}{\partial x} (5 x^{2} - 3 x y + x yz)

What is (\partial)/(\partial x)(e^{xy}cos(bx+c)) ?
The answer to (\partial)/(\partial x)(e^{xy}cos(bx+c)) is e^{xy}ycos(bx+c)-be^{xy}sin(bx+c)