Question

Partial derivative of 1+2x+3xy+4xyz with respect to x​

Answer 1

Partial Derivatives

The concept of partial derivatives comes in when we have more than one variable in a function.

When we take the partial derivative of a function with respect to some variable, we consider all the other variables as constant and then go ahead just like we were taking a normal derivative.

For example, here, we have a function in terms of x, y and z. Let's name this function as f(x,y,z).

$\sf f(x,y,z) = 1+2x+3xy+4xyz$

Just like a normal derivative is denoted by $\sf d$, we denote a partial derivative by $\partial$ [read as "partial"].

So, the partial derivative of f with respect to x [denoted as $\sf \frac{\partial f(x,y,z)}{\partial x}$] just means a derivative of f with respect to x as if y and z were constant.

Hence, we have:

$\displaystyle \sf f(x,y,z) = 1+2x+3xy+4xyz \\\\\\ \sf \implies \frac{\partial f(x,y,z)}{\partial x} = 0 + 2 + 3y + 4yz \\\\\\ \implies \boxed{\frac{\partial f(x,y,z)}{\partial x} = 2+3y+4yz}$

Answer 2

Partial Derivatives

The concept of partial derivatives comes in when we have more than one variable in a function.

When we take the partial derivative of a function with respect to some variable, we consider all the other variables as constant and then go ahead just like we were taking a normal derivative.

For example, here, we have a function in terms of x, y and z. Let's name this function as f(x,y,z).

$\sf f(x,y,z) = 1+2x+3xy+4xyz$

Just like a normal derivative is denoted by $\sf d$, we denote a partial derivative by $\partial$ [read as "partial"].

So, the partial derivative of f with respect to x [denoted as $\sf \frac{\partial f(x,y,z)}{\partial x}$] just means a derivative of f with respect to x as if y and z were constant.

Hence, we have:

$\displaystyle \sf f(x,y,z) = 1+2x+3xy+4xyz \\\\\\ \sf \implies \frac{\partial f(x,y,z)}{\partial x} = 0 + 2 + 3y + 4yz \\\\\\ \implies \boxed{\frac{\partial f(x,y,z)}{\partial x} = 2+3y+4yz}$