Partial differentiation must be applied when we want to differentiate a function of two or more variables.
The differential ofwith respect toisIf instead we are given to differentiatewith respect tothen we treatas a constant, likeobtainingThe partial derivative ofwith respect towhenis a function of several variables is written and the partial derivative with respect tois writtenThe partial derivatives give the gradients of the tangent lines in the direction of the coordinate axes.
Is the gradient of the tangent in the- direction andis the gradient of the tangent in the– direction.
Example: Find the partial derivative ofwith respect to a)and b)
a)We differentiate using the product rule.
b)Regarding x as a constant gives
The derivative ofisso by the chain rule the derivative ofis given by
and the derivative of