What is (\partial)/(\partial x)(e^{2x}*cos(x+y)) ?

The answer to (\partial)/(\partial x)(e^{2x}*cos(x+y)) is 2e^{2x}cos(x+y)-e^{2x}sin(x+y)

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(\partial)/(\partial x)(e^{2x}*cos(x+y))

\frac{\partial}{\partial x} (e^{2 x} \cdot cos (x + y))

2 e^{2 x} cos (x + y) - e^{2 x} sin (x + y)

Solution steps

\frac{\partial}{\partial x} (e^{2 x} cos (x + y))

Treat

y

as a constant

Apply the Product Rule:

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

f = e^{2 x}, g = cos (x + y)

= \frac{\partial}{\partial x} (e^{2 x}) cos (x + y) + \frac{\partial}{\partial x} (cos (x + y)) e^{2 x}

\frac{\partial}{\partial x} (e^{2 x}) = e^{2 x} \cdot 2

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

= e^{2 x} \cdot 2 cos (x + y) + (- sin (x + y)) e^{2 x}

Simplify

= 2 e^{2 x} cos (x + y) - e^{2 x} sin (x + y)

y = ∣ 2 x ∣, y = x^{2} - 3

\frac{d}{d x} (e^{- 2 x} (sin (3 x) + cos (3 x)))

\frac{d}{d x} (- 8 sin (4 x))

x \to 3 + lim (x)

\int_{0}^{2} x^{2} e^{x^{3}} d x

What is (\partial)/(\partial x)(e^{2x}*cos(x+y)) ?
The answer to (\partial)/(\partial x)(e^{2x}*cos(x+y)) is 2e^{2x}cos(x+y)-e^{2x}sin(x+y)