How do you differentiate x e x 2 + y 2
How do you differentiate
Answer & Explanation
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
So, deriving with respect to x, and using the product rule where and , we get
Where the derivative of has been calculated using the chain rule, which states that , where , and
The derivative with respect to y is easier, since the only factor to differentiate is , while the others depend only on x and are thus to be consideblack as constant. So, we have
The gradient is thus the vector