The Concept of Differentiation
The derivative of a function tells the rate of change of w.r.t. . More specifically, the derivative measures the sensitivity to change of a quantity , which is determined by another quantity . is known as the independent variable and is known as the dependent variable.
Formally, the derivative is defined by the limit
It finds its use in wide range of areas and is also linked to integration, which leads to the fundamental theorem of integral calculus.
Successive Differentiation in Dynamics
Let the displacement , velocity and acceleration be functions of time . By definition,
The jerk is defined as the rate of change of acceleration. So,
Thus, are obtained by differentiating successively.
Let be a function of only; . Then
is the first derivative,
is the second derivative,
is the third derivative.
In general, is the ‘n’th derivative}.
Infinitely Differentiable Functions
There are certain functions, whose derivatives exist for all values of . These are known as infinitely differentiable functions. We are going to study these functions only.
th Derivatives of Standard Functions
I) Trigonometric Formulas
II) Euler’s Identity
III) Splitting into Partial Fractions
To get , we solve , put the value in , except . . Similarly, to get , we solve , and put the value in , except . .
These fractions can be easily differentiated by using where .
IV) Combinations of objects, taken at a time, :
Leibniz Rule (Leibnitz Rule)
The rule is due to Leibniz, who invented calculus independently of Newton. His rule is useful for finding th derivative of product of 2 functions. The rule says, if , then,
HINT FOR MCQs
In any problem asking for th derivative of a function, get the first derivative of the function. Put in the options. See which of the options matches with the first derivative. If two or more options match, get the second derivative, put in the options and check.
Example : th derivative of is :
I) , II) , III) , IV)
Put , thus , i.e. , so first derivative will be . Option .