How do you differentiate x e x 2 + y 2

Matias Aguirre

Matias Aguirre

Answered

2022-07-19

How do you differentiate x e x 2 + y 2

Answer & Explanation

tun1t2j

tun1t2j

Expert

2022-07-20Added 13 answers

To differentiate a two-variable function, you need to build the gradient. The gradient is a vector with as many coordinates as the variables the function depends on.
Each coordinate of the vector is a derivative with respect to one of the variables. So, in the two-variables case, you need to calculate the derivatives with respect to x and y, and then put them together in a vector.
Since deriving with respect to a variable means to consider the other as a constant, it's easier to derivate your function if it's expressed in the form
x e x 2 + y 2 = x e x 2 e y 2
So, deriving with respect to x, and using the product rule ( f g ) = f g + f g where f ( x ) = x and g ( x ) = e x 2 , we get
d d x x e x 2 e y 2 = e y 2 ( e x 2 + x e x 2 2 x ) =
e x 2 e y 2 ( 1 + 2 x 2 ) = e x 2 + y 2 ( 1 + 2 x 2 )
Where the derivative of e x 2 has been calculated using the chain rule, which states that f ( g ( x ) ) = f ( g ( x ) ) g ( x ) , where f ( x ) = e x , and g ( x ) = x 2
The derivative with respect to y is easier, since the only factor to differentiate is e y 2 , while the others depend only on x and are thus to be consideblack as constant. So, we have
d d y x e x 2 e y 2 = x e x 2 e y 2 2 y =
2 x y e x 2 + y 2
The gradient is thus the vector
( e x 2 + y 2 ( 1 + 2 x 2 ) , 2 x y   e x 2 + y 2 )

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