Question

How does a partial derivative differ from an ordinary derivative?

Answer 1

In this question we are asked to find how a partial derivative differs from an ordinary derivative. To answer this question we have to define a partial derivative and an ordinary derivative. And also we need to give some examples for the clear explanation.

Complete step-by-step solution:
In partial differentiation we used to differentiate mathematical functions having more than one variable in the expression.
When we are taking the partial derivative with respect to x, we will treat the variable y as a constant.
This means an expression like${y^2}$ just looks like some constant power two, which is again a constant.
In ordinary differentiation, we find derivatives with respect to only one variable, as function contains only one variable.
So partial differentiation is more general than ordinary differentiation.
Example for partial derivative:
Consider $f(x,y,z) = 4{x^2}y + 2y + 2z$ differentiate $f$ partially with respective $x$
$\Rightarrow {f'} = 8xy$
Here we differentiate only $x$ and other variables like y and z are considered as constants. So their derivatives are zero.
In ordinary differentiation, all the variables are differentiated with respect to the considered variables.
Example for ordinary derivative:
Let’s considered the above function
Ordinary differentiation with respect to $x$ is ${f'} = 8x\dfrac{{dy}}{{dx}} + 2\dfrac{{dy}}{{dx}} + 2\dfrac{{dz}}{{dx}}$
$\Rightarrow {f'} = (8x + 2)\dfrac{{dy}}{{dx}} + 2\dfrac{{dz}}{{dx}}$
In this way partial differentiation and ordinary differentiation differed from each other.

Note: In partial differentiation other than the differentiating variable other variables are treated as constants. If the differentiating variables are combined with the constant variable then after differentiation the whole term won’t become zero. In this part the constant term remains constant. But in ordinary differentiation the constant term also differentiated with respect to the differentiating variable.