22. Differentiation - Revision points 1

4. Fundamental rules for differentiation

1. d/dx of constant = 0

2. d/dx of (c.f(x)) = c d/dx of f(x)

3. d/dx of (f(x) ±g(x)) = d/dx of f(x) ± d/dx of (g(x))

4. Product rule d/dx of uv = u*dv/dx + v*dv/dx

5. quotient rule d/dx of u/v = [v*du/dx – u*dv/dx]/v²

5. Relation between dy/dx and dx/dy

dy/dx = 1/{dx/dy)

6. Differentiation of implicit functions

If variables are given as f(x,y) = 0 and if it is not possible to find y as a function of x in the form y = ф(x), then y is said be an implicit function of x.

To find dy/dx in such a case, differentiate both sides of the equation with respect to x, by writing the derivative of g(y) w.r.t. to x as (dg/dy)*(dy/dx).


7. Logarithmic differentiation

To find derivatives of the functions of the form [f(x)] ^g(x)

Procedure is:
Let y = [f(x)] ^g(x)

Take logarithms on both sides

Log y = g(x)*log [f(x)]

Differentiate w.r.t. x
(1/y)*dy/dx = g(x)*(1/f(x))*d(f(x))/dx + log [f(x)]*d[g(x)]/dx

Therefore dy/dx = (1/y)*[ g(x)*(1/f(x))*d(f(x))/dx + log [f(x)]*d[g(x)]/dx]


8. Differentiation of parametric form

If x = f(t) and y = g(t) are given and we have to find dy/dx

Then, first find dy/dt and dx/dt

dy/dx will be obtained as (dy/dt)/(dx/dt)

9. Differentiation of a function with respect to another function

u = f(x) and v = g(x) be two functions. To find the derivative of f(x) with respect of g(x) or du/dv use the formula

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

10. Higher order derivatives

Derivative of y w.r.t. x = dy/dx

Derivative of dy/dx w.r.t. x = d²y/dx²

and so on.

The alternative notations of higher order derivatives are

dy/dx, d²y/dx²

y₁, y₂

y’, y’’

Dy, D²y

f’(x), f’’(x)