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DIFFERENTIATION

WHAT IS DIFFERENTIABILITY

A function f(x) is said to be differentiable at x = a if limit _xèa {f(x) - f(a)}/(x - a) exists. If this limit exists, then it is written as

f’(a) = limit _xèa{f(x)-f(a)}/(x - a)

f’(a) is the derivative of f(x) at x = a.

In the above limit, if we put x - a = t, then

f’ (a) = limit_tè0 {f(t+a)-f(a)}/t

Which is another form of the definition of differentiability.

Note: Differentiability always implies continuity. i.e. if a function is differentiable at a point, it is continuous at that point. But if a function is continuous at a point it may not always be differentiable at that point.

The sum, difference, product and quotient of two continuous and differentiable functions is continuous and differentiable.

The functions a^x, sinx, cosx, logx and all polynomials are all differentiable everywhere in their domains.

Using the Definition to Compute the Derivative

We have seen in the previous page how the derivative is defined: For a function f(x), its derivative at x=a is defined by

Fop the function f(x) = x². We have

thus

which means f ‘(a) = 2a.

What about the derivative of f(x) = xⁿ. Calculations, using the binomial expansion for (x+y)ⁿ gives

What is differentiation?

The process of finding the derivative of a function in differential calculus. If y = f(x) then the derivative of y can be written as dy/dx or f’(x), this is equal to the limit as Dx tends to 0 of [f (x + Dx) - f(x)] /Dx. In general if y = xⁿ then dy/dx = nx^n-1. On a graph, the derivative of dy/dx is the gradient of the tangent to the curve at the point x.

Differential calculus treats a continuously varying quantity as if it consisted of an infinitely large number of infinitely small changes. For example the velocity v of a body at a particular instant can be regarded as the infinitesimal distance ds it travels in the vanishingly small time dt; the instantaneous velocity is then ds/dt. This term ds/dt is called the derivative of s with respect to s.

FUNDAMENTAL THEOREMS

1. DERIVATIVE OF A SUM: If y = u + v, then the derivative of y is d(u+v)/dx which can be written as

   dy/ dx = du/dx + dv/

2. DERIVATIVE OF A DIFFERENCE: If y = u - v, then the derivative of y is d(u-v)/dx which can be written as

   dy/ dx = du/dx - dv/d

3. DERIVATIVE OF A PRODUCT: If y = uv, then the derivative of y is d(uv)/dx which can be written as

   dy/ dx = udv/dx + vdu/

4. DERIVATIVE OF A QUOTIENT: If y = u/v, then the derivative of y is d(u/v)/dx which can be written as

   dy/ dx = [vdu/dx - udv/dx] v

5. FUNCTION OF A FUNCTION: If y = f(u) and u = g(x) then the derivative of y is which can be written as,

   dy/ dx = (dy/du)(du/dx

6. DIFFERENTIATION OF A FUNCTION WITH RESPECT TO ANOTHER FUNCTION: If y = f(v) and x = g(v) then the derivative of y with respect to x is which can be written as,

dy/ dx = (dy/du)(dv/dx

DERIVATIVE OF AN IMPLICIT FUNCTION

In order to find the derivative of an implicit function F(x,y) = 0 we differentiate F with respect to x considering y as a function of x and then solve the resulting equation. Suppose we have to differentiate x³ - y³ - 3xy = 0, then we proceed as follows; we first differentiate wrt x and get 3x² + 3y² - 3(y + xdy/dx) = 0. Then we solve the equation dy/dx and then get dy/dx = -(x² - y)/ (y²-x).

SOME BASIC FORMULAE

1. y = xⁿ then dy/dx = nx^n-1
2. y = uⁿ then dy/dx = nu^n-1du/dx
3. y = x^1/2 then dy/dx = 1/2x^1/2
4. Logarithmic functions

• y = log x then dy/dx = 1/x

1. Exponential functions

• If y = e^x then dy/dx = e^x

1. y = a^x then dy /dx = a^x log a
2. Trigonometric functions

• y = sin x then dy/dx = cos x
• y = cos x then dy/dx = -sin x
• y = tan x then dy/dx = sec² x
• y = cot x then dy/dx = cosec² x
• y = sec x then dy/dx = sec x tan x
• y = cosec x then dy/dx = -cosec x cot x

1. Inverse trigonometric functions

• y = sin^-1 x then dy/dx = 1/(1-x²)^1/2
• y = cos^-1 x then dy/dx = -1/(1-x²)^1/2
• y = tan^-1 x then dy/dx = 1/(1+x²)
• y = cot^-1 x then dy/dx = -1/(1+x²)
• Chain Rule
  The formulais known as the Chain Rule formula. It may be rewritten as