What is (\partial)/(\partial x)(xy(sin((x^5)+y^5))) ?

The answer to (\partial)/(\partial x)(xy(sin((x^5)+y^5))) is y(sin(x^5+y^5)+5x^5cos(x^5+y^5))

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(\partial)/(\partial x)(xy(sin((x^5)+y^5)))

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

y (sin (x^{5} + y^{5}) + 5 x^{5} cos (x^{5} + y^{5}))

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = x, g = sin (x^{5} + y^{5})

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

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

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

= y (1 \cdot sin (x^{5} + y^{5}) + cos (x^{5} + y^{5}) \cdot 5 x^{4} x)

Simplify

y (1 \cdot sin (x^{5} + y^{5}) + cos (x^{5} + y^{5}) \cdot 5 x^{4} x) : y (sin (x^{5} + y^{5}) + 5 x^{5} cos (x^{5} + y^{5}))

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

\int_{- 1}^{2} \frac{1}{x ^{2}} d x

\frac{d}{d x} (∣ x - 2 ∣)

e^{(\frac{3}{x})}

\frac{d}{d x} ((x^{5} - 7)^{2})

\int \frac{sin ( ln ( x ) )}{x} d x

What is (\partial)/(\partial x)(xy(sin((x^5)+y^5))) ?
The answer to (\partial)/(\partial x)(xy(sin((x^5)+y^5))) is y(sin(x^5+y^5)+5x^5cos(x^5+y^5))