# Derivatives Index

## Derivatives of Logarithmic and Exponential functions.

Exponential functions are the function which are defined in the form of: f(x)=ax , where a is a constant and “x” is a variable. The function “f(x) = ax“ is called an exponential function in base “a”. The logarithmic functions are the inverse function of exponential function. Or , if ” y = f(x) = ax ” then , x=f-1(y)  is called the logarithmic function and is denoted by y= log a x , which is called the logarithmic function in base “a”. And , The natural exponential is the exponential...

## Derivatives of inverse trigonometric functions

Inverse trigonometric functions  are the  inverse of trigonometric functions . For example if, y = sinx  then the inverse function of y = sinx is , is denoted by: x=sin-1y and is called inverse sin function. You should note that: doesn’t means instead “y = sin -1 x” is the inverse function of “x = sin y” Derivatives of Inverse Trigonometric Functions: The derivatives of inverse sine , inverse cos , inverse tan , inverse csc , inverse sec , inverse cot functions are given below: Derivative of inverse sin...

## Derivatives of Trigonometric functions.

$Derivatives of Trigonometric functions.$

As you know, The functions SINE x(sin x) , CO-SECANT x(cos x) , TANGENT x(tan x), CO-SECANT x(csc x), SECANT x(sec x) and COTANGENT x(cot x)  are called trigonometrical functions. You can learn more about these functions by searching about it in the search box above. We are going to learn and prove ,what are the Derivatives of these Trigonometrical Functions , here. a> Derivative of sin x: The derivative of sin x is: Proof: Ley y=SIN x and let this be equation (i) and let  be a small increment in x and be the corresponding small...

## Infinitesimals and Differentials.

In the chain rule we have come across the following relation: Which gives us impression that is a fraction of “dy” and “dx” And in right hand side of above relation we can cancel “du” in the numerator and denominator to get of the left hand side. Actually in the early days Leibnitz and others treated as a fraction of “dy” and “dx” or delta y and delta x which are called infinitesimals or infinitely small numbers having basic properties of the number zero. In fact this is the...

## Derivative of implicit functions.

Finding derivative of a explicit functions is easy using the differentiation techniques but It is difficult to find the derivative of implicit  functions so we use the Implicit Differentiation technique to find the derivative of Implicit functions. The Implicit Differentiation technique make use of the Chain Rule and the Sum Rule. Example of Implicit Differentiation: let us Differentiate y with respect to x in this Implicit function: 4x3-3y2=21 Differentiating both side of equation with respect to x , and using The Sum Rule: Now using...

## Second and higher derivatives.

Think of a function y=f(x) , and let y=f(x) be a differentiable function. Then you can differentiate the function f(x) with respect to x or find derivative of f(x)= which is called the first derivative of the function f(x). Let the first derivative of the function be “y” and now if you again differentiate y with respect to “x” then the result is called second derivative of function f(x) you can again find it’s derivative and that’s called the third derivative of function f(x) and similarly you can...

## The Chain Rule.

Chain Rule is one of the Techniques of Differentiation. The Chain Rule states that: If v(u) is a Function of “u” and u(x) is a function of “x” then  the Derivative of function “v” with respect to “x” exists which  is equal to the product of derivative of function “v” with respect “u” and derivative of function “u” with respect to “x”. Mathematically it can be written as: If,  “v” is a function of “u” and...

## The Quotient Rule.

Quotient Rule is one of the Techniques of Differentiation. The Quotient Rule states that: The Derivative of  a Function “f(x)” divided by another function “g(x)” is the difference between the second function multiplied by derivative of first function and first function multiplied by derivative of second function whole divided by square of  the second function. Mathematically we can write: Proof of Quotient Rule: If, and “a” is a fixed point then, Use of Quotient Rule: Find the...

## The Power Rule.

Power Rule is one of the Techniques of Differentiation. The Power Rule states that: The Derivative of  a Function raised to n’th power is the Product of n , the function raised to the power (n-1)’th and derivative of the function raised to first power. Mathematically we can write: We can also further synthesize this relation and prove the rule for Derivative of general Polynomial Function. which states: Proof Of Power Rule for Natural N’th Power: The power rule is valid for any rational “n”(eg: 4/5)...

## The Product Rule.

Product Rule is one of the Techniques of Differentiation. The Product Rule states that: The Derivative of  product of two Function is the Sum of Derivative of first function multiplied by second function and first function multiplied by derivative of second function. Mathematically we can write: Proof of Product Rule: If, and ‘a’ is a fixed point Then, Use of Product Rule: Find the derivative of:...