# Implicit Functions

A function f (xy) = 0 is said to be an implicit function, if y cannot be directly defined as a function of x.

In such a case, we will differentiate both sides of this equation w.r.t x, collect the terms containing
on one side; transfer other terms to the other side and divide by the coefficient ofto get its value.

Example
Solution
Given

# Parametric Equation

If both x and y are functions of a given independent term ti.e.x = f(t) and y = g(t), then the equations containing this x and y are called parametric equations.

The differential of such an equation is given by

Example
Solution

# Logarithmic Differentiation

When a function is expressed in any of the following forms, its derivative can be obtained by taking the logarithm of the function and then differentiating it.
• A product of a number of functions
• When a function is raised to some exponent which is also a function
• A number of functions are divided

This method of differentiation is known as logarithmic differentiation.

Example
Solution
Given

Differentiating both sides, we get

# Higher Order Derivatives

Let y = f (x), be a function of xis called the first derivative of y with respect to x.

The derivative of f â€²(x) is called the second derivative of y with respect to x,

i.e.

Example
Solution

Similarly the derivative of f â€²â€²(x) is called the third derivative of y with respect to x.
The nth derivative of y with respect to x is given by

# Geometric Interpretation of the Derivative

Let y = f (x) be a curve as shown below.

Let P(xy) and  be two neighbouring points. Join these two points and extend it to meet the x axis at point M.

Slope of PQ is given by

As Q approaches P, the line QPM becomes the tangent to the curve at P and the angle q approaches y.

Hence, the derivative of y with respect to x is the slope of the tangent to the curve y = f (x) at the point P(xy).