Question

Apply a chain rule to calculate \frac{\partial a}{\partial x} ∂x
∂a
​ where a(x, y) = sin(xy)\cdot e^x a(x,y)=sin(xy)⋅e x
.
Here is an example of the syntax: sin(x*y)*exp(x), more info here

Answer 1

Given :  a(x , y)  = sin(xy) eˣ

To find :  ∂a /∂x

Solution:

a(x , y)  = sin(xy) eˣ

Applying product rule ( f .g )'  = f'g  + fg'

f  = Sin(xy)   g  = eˣ

∂a /∂x   = ( ∂Sin(xy)  /∂x .  )eˣ  + Sin(xy)  ∂eˣ/∂x

now   ∂eˣ/∂x  =   ∂eˣ/∂x

&  ∂Sin(xy)/∂x   =  Cos(xy) y

=> ∂a /∂x  =  Cos(xy) y eˣ  +  Sin(xy) eˣ

=>  ∂a /∂x  =   eˣ ( yCos(xy)  + Sin(xy) )

∂a /∂x  =   eˣ ( yCos(xy)  + Sin(xy) )

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