What is (\partial)/(\partial y)(cos(x+e^y)e^y) ?

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

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(\partial)/(\partial y)(cos(x+e^y)e^y)

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

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

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

x \to 7 - lim (\frac{( x - 9 )}{x - 7})

t^{5} e^{3 t}

\int \frac{4}{x ^{2} - 8 x} d x

y^{^{''}} + 16 y = 2 sin (4 x)

y^{^{''}} - k^{2} y = 0

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