# Implicit Functions

A function*f*(

*x*,

*y*) = 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 containingon one side; transfer other terms to the other side and divide by the coefficient ofto get its value.

# Parametric Equation

If both*x*and

*y*are functions of a given independent term

*t*,

*i.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

# 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.

Differentiating both sides, we get

# Higher Order Derivatives

Let*y*=

*f*(

*x*), be a function of

*x*. is 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.*,

Similarly the derivative of

*f*â€²â€²(

*x*) is called the third derivative of

*y*with respect to

*x*.

The

*n*th 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*(*x*, *y*) 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*(*x*, *y*).