Partial Derivative Rules

The calculus generally has multiple variables, but it is very similar to the single-variable calculus and is only applied to multiple variables one at a time. We get a partial derivative when we hold all but one of the independent variables of a function constant and differentiate concerning that one variable. The use of partial derivatives is very extensive nowadays as many mathematical calculations have become calculus-based.

The partial derivatives are very different from ordinary derivatives and we use the symbol ∂ rather than the d previously used for ordinary derivatives. The partial derivative symbol is pronounced differently as “del”, “partial dee”, “doh”, or “dabba”.

Partial Derivative

The partial derivative of f(x,y) with respect to x at the point (x, y) is ∂f/∂x (x,y) = lim h→0 f(x +h,y) − f(x, y)h , provided that the limit exists.

The slope of the curve is denoted by,

z = f (x, y0) at the point A(x0, y0, f (x0, y0)) in the plane y = y0 is the value of the partial derivative of the function f with respect to x at (x0, y0). 

The line tangent to the curve at A is the line in the plane y = y0 that passes through A along this slope. The partial derivative ∂f /∂x at (x0, y0) gives the rate of change off concerning x when y is held fixed at the value y0.

Chain rule for partial derivatives:

• If z = f(x, y) and x and y are functions of t (x = x(t) and y = y(t)) then z is ultimately a function of t only and dz/dt = ∂z/∂x*dx/dt + ∂z/∂y*dy/dt.
• If w = f(x, y, z) and x = x(t), y = y(t), z = z(t) then w is ultimately a function of t only and dw/dt = ∂w/∂x*dx/dt + ∂w/∂y*dy/dt + ∂w/∂z*dz/dt.

Transformation to polars:

• Let u = u(x, y) be a function of x and y. Let x = r cos θ, y = r sin θ, then 

∂u/∂r = cos θ*∂u/∂x + sin θ*∂u/∂y. 

Properties of partial derivatives:

1. If f (x, y) and its partial derivatives fx, fy, fxy, and fyx are defined throughout in an open bracket region containing a point (a, b), and if here all are continuous at (a, b), then fxy (a, b) = fyx (a, b) always.
2. If F(x, y) is differentiable, the equation F(x, y) = 0 defines y as a differentiable function of x. Then at any point where Fy is not equal to 0, dy dx = − Fx/Fy.
3. If the point is (a, b) and a nearby point is (a + h, b + k), the distance between them is √(h2 + k2).
4.  A function f (x, y) can have partial derivatives for both x and y at a point without the function being continuous there. This is different from the functions of a single variable, where the existence of a derivative implies continuity.

Conclusion

Partial Derivatives are the functions used to calculate the rate of change. They can be in single variables or multiple variables. The partial derivatives are very different from ordinary derivatives and the function z = f(x, y) of two independent variables x and y extend the concept of ordinary derivative of the function of one variable to the function z = f(x, y) by keeping y constant while taking derivative for x and keeping x constant while taking derivative for y. we use the symbol ∂ rather than the d previously used for ordinary derivatives.