Differentiation

Concept of Derivatives
Derivatives find the amount of change in y, with an infinitesimally small change in x.

Operations
1)

Composite Functions
g(f(x)) can be differentiated as d f(x)/dx. d (g(x))/dx

Inverse Functions
Inv (A) : Substitution of the variables to simplify the equation.

Inv (B) : Substitution using pythagorean triplets of a triangle.

Logarithmic Differentiation
1) This method is used when a variable is raised to a variable.

2) Take log on both sides.

3) Differentiate w.r.t. x

Log(A) : Differentiating logs on both sides. (Used with multiplication and Division)

log (B) : Differentiating logs separately and then addition / subtraction (Used with Addition and Subtraction)

Explicit & Implicit Functions
Explicit Function : y = 3x2 + 2x + 5

Implicit Function : y2 + x2 + xy = 0

To differentiate implicit functions, differentiate the entire equation w.r.t. x and then bring dy/dx terms together.

Parametric Functions
Parametric functions, are those functions which can be expressed using a common variable. We first differentiate both functions w.r.t. the common variable. Then we divide their differential values, to find their differential w.r.t. each other.

Para (A) : Differentiating separately with respect to the common variable and then dividing them.

Para (B) : Differentiate separately and then divide dx/dy

Higher Order Differential
Higher Order differentiation deals with differentiating a function multiple times.

Differentiability and Continuity
If a function is differentiable at a given point, then it is continuous at that point.

If a function is continuous at a point, it is not necessary that it is differentiable.

1) Rate Measurer
When an equation is differentiated with respect to time, we can measure the rat of change.

2) Tangents and Normals
Slope of tangent = dy/dx = tanθ

Slope of Normal = -dx/dy = -1/tanθ

Equation of Tangent :  y - y1 = m ( x - x1)            m = dy/dx

Equation of Normal : y - y1 = m ( x - x1)                m = - dx/dy

length of Tangent = y1 sqrt (1 + (dx/dy)2)

length of Normal = y1 sqrt (1 + (dy/dx)2)

length of sub-tangent = | y1(dx/dy) |

length of sub-normal = | y1(dy/dx) |

Tips &Tricks
1) Implicit