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Popular Calculus >

derivative of e^ycos(xy)

Solution

\frac{d}{d x} (e^{y} cos (x y))

Solution

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

Solution steps

\frac{d}{d x} (e^{y} cos (x y))

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

Popular Examples

tangent of y=(x^3-25x)^8,(-5,0)

t an g e n t y = (x^{3} - 25 x)^{8}, (- 5, 0)

derivative of (x^4-3x^2/((x^2-1)^2))

\frac{d}{d x} (\frac{x ^{4} - 3 x ^{2}}{( x ^{2} - 1 ) ^{2}})

area x=((y-2)^2)/2 ,y=6-x

a re a x = \frac{( y - 2 ) ^{2}}{2}, y = 6 - x

tangent of f(x)= x/((3x-2)^6),\at x=1

t an g e n t f (x) = \frac{x}{( 3 x - 2 ) ^{6}}, at x = 1

integral of x^5e^{x^3-7}

\int x^{5} e^{x^{3} - 7} d x

Frequently Asked Questions (FAQ)

What is the derivative of e^ycos(xy) ?
The derivative of e^ycos(xy) is e^ycos(xy)(dy)/(dx)-e^ysin(xy)(y+x(dy)/(dx))
What is the first derivative of e^ycos(xy) ?
The first derivative of e^ycos(xy) is e^ycos(xy)(dy)/(dx)-e^ysin(xy)(y+x(dy)/(dx))