Geometrical Interpretation of Partial Derivative

As we know, for single variable functions, the derivative is computed as the slope of the tangent passing through the curve. Similarly, we can understand the geometric interpretation of a partial derivative of a multivariable function.

Consider a function of two variables, z = f(x,y) on the 3-D plane and let a plane y=b pass through the curve f(x,y). 

Now, we draw another curve f(x,b) lying on z that is perpendicular to the plane y=b. Consider two arbitrary points P,R on this curve and draw the secant passing through these points. 

The slope of this secant is calculated using the first principles as follows :

[Tex]m = \frac{Δz}{Δx} = \frac{f(x+Δx,b)-f(a,b)}{Δx}[/Tex]

As the two points move closer to each other, the difference Δx approaches 0 and we calculate this in the form of the limit : [Tex]\lim_{Δx\to0} \frac{Δz}{Δx} = \frac{f(a+Δx,b)-f(a,b)}{Δx}[/Tex]

This limit is the partial derivative of ‘z’ with respect to ‘x’ by treating ‘y’ as constant i.e. 

[Tex]\frac{\partial z}{\partial x} = \frac{f(a+Δx,b)-f(a,b)}{Δx}[/Tex]

Partial Derivatives in Engineering Mathematics: A function is like a machine that takes some input and gives a single output. For example, y = f(x) is a function in ‘x’. Here, we say ‘x’ is the independent variable and ‘y’ is the dependent variable as the value of ‘y’ depends on ‘x’. 

Some examples of functions are:

1. f(x) = x² + 3 is an algebraic function.
2. e^x is the exponential function.
3. sin(x), cos(x), tan(x),…etc. are all trigonometric functions.

Now, all these functions are functions of a single variable, i.e. there is only one independent variable.