# Differentiation of Functions in Parametric Form

Sometimes x and y are given as functions of a single variable, e.g., x = Ï†(t), y = Ïˆ(t) are two functions and t is a variable. In such a case x and y are called parametric functions or parametric equations and t is called the parameter. To find dy/dx in case of parametric functions, we first obtain the relationship between x and y by eliminating the parameter t and then we differentiable it with respect to x. But every time it is not convenient to eliminate the parameter. Therefore, dy/dx can also be obtained by the following formula:

=

# Differentiation using logarithm

If y =  or y = f1(xâ‹… f2(xâ‹… f3(x) ... or
y

then it is convenient to take the logarithm of the function first and then differentiate.

Note: Write y =  and differentiate easily or if y = [f(x)] g(x), then dy/dx = differential of y treating f(x) as constant + differential of y treating g(x) as constant.

For example, if y = , then we can find  by the following steps:
= (differential of y keeping base sin x as constant) + (differential of y keeping power log case x as constant)

# Differentiation of one function w.r.t. other function

Let u = f(x) and v = g(x) be two functions of x. Then to find the derivative of f(x) w.r.t. g(x), i.e., to find du/dv we use the following formula .

Thus, to find the derivative of f(x) w.r.t. g(x), we first differentiate both w.r.t. x. and then divide the derivative of f(x) wrt x by the derivative of g(x) w.r.t. x

# Differentiation of determinants

To differentiate a determinant, we differentiate one row (or column) at a time, keeping others unchanged.

For example, if
Î”(x) =
then  {Î”(x)} =
Also, {Î”(x)} =

Similar results hold for the differentiation of determinants of higher order.

# Higher-order derivatives

If y = y(x), then dy/dx, the derivative of y with respect to x, is itself, in general, a function of x and can be differentiated again. We call dy/dx as the first-order derivative of y with respect to x and the derivatives of dy/dx w.r.t. x as the second-order derivative of y w.r.t. x and will be denoted by d2y/dx2. Similarly, the derivative of d2y/dx2 w.r.t. x will be termed as the third-order derivative of y w.r.t. x and will be denoted by d3y/dx3 and so on. The nth-order derivative of y w.r.t. x will be denoted by dny/dxn.

If y = f(x), then the other alternative notations for

are
y1y2y3, â€¦, yn
yâ€², yâ€²â€², yâ€²â€²â€², â€¦, y(n)
fâ€²(x), fâ€²â€²(x), fâ€²â€²â€²(x), â€¦, fn(x)

The values of these derivatives at x = a are denoted by
yn(a), yn(a), Dny(a), fn(a) or