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Differentiation of Functions in Parametric Form

Sometimes x and y are given as functions of a single variable, e.g., x = φ(t), y = ψ(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:
85010.png = 85004.png

Differentiation using logarithm

If y = 84998.png or y = f1(x f2(x f3(x) ... or
y 84992.png
then it is convenient to take the logarithm of the function first and then differentiate.


Note: Write y = 85626.png 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 = 84961.png, then we can find 84955.png by the following steps:
84949.png = (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 84930.png.
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
Δ(x) = 84924.png
then 84920.png {Δ(x)} = 84914.png
Also, 84908.png{Δ(x)} = 84902.png
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
84853.png are
y1y2y3, …, yn
y′, y′′, y′′′, …, y(n)
f′(x), f′′(x), f′′′(x), …, fn(x)
The values of these derivatives at x = a are denoted by
yn(a), yn(a), Dny(a), fn(a) or 84847.png

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