What is (\partial)/(\partial x)(sin(x)*sin(x+y)) ?

The answer to (\partial)/(\partial x)(sin(x)*sin(x+y)) is sin(2x+y)

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(\partial)/(\partial x)(sin(x)*sin(x+y))

\frac{\partial}{\partial x} (sin (x) \cdot sin (x + y))

sin (2 x + y)

Solution steps

\frac{\partial}{\partial x} (sin (x) sin (x + y))

Treat

y

as a constant

Apply the Product Rule:

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

f = sin (x), g = sin (x + y)

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

\frac{\partial}{\partial x} (sin (x)) = cos (x)

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

= cos (x) sin (x + y) + cos (x + y) sin (x)

Simplify

cos (x) sin (x + y) + cos (x + y) sin (x) : sin (2 x + y)

= sin (2 x + y)

\int sin^{5} (8 x) d x

f (x) = 4 x^{2}

\frac{d}{d x} (\frac{x ^{2} - 1}{e ^{x}})

\frac{d}{d x} (- 9 sin (x))

x \to - 1 lim (3 x - 2)

What is (\partial)/(\partial x)(sin(x)*sin(x+y)) ?
The answer to (\partial)/(\partial x)(sin(x)*sin(x+y)) is sin(2x+y)