Introduction to Differential Calculus
Last Updated :
11 Dec, 2024
Differential calculus is a branch of calculus that deals with the study of rates of change of functions and the behaviour of these functions in response to infinitesimal changes in their independent variables.
Some of the prerequisites for Differential Calculus include:
What is Limit?
For a function y = f(x), then limit x approaches a for function y = f(x) represents the value function approaches when we approach the input value x = a. In simple words, the limit of any function at a given point tells us about its behaviour at and around the point of consideration. It is given as lim x⇝a f(x). Limit is unique in nature i.e. for x tends to a, there can't be two values of f(x).
Left Hand and Right Hand Limit
| Left-Hand Limit | \lim_{x \to a^{-}}f(x)=\lim_{h \to 0^{-}}f(a-h) |
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| Right-Hand Limit | \lim_{x \to a^{+}}f(x)=\lim_{h \to 0^{+}}f(a+h) |
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Existence of Limit
For Existence of \bold{\lim_{x \to a}f(x)} ,
- Both \lim_{x \to a^{-}}f(x) and \lim_{x \to a^{+}}f(x) exists, and
- \lim_{x \to a^{-}}f(x) = \lim_{x \to a^{+}}f(x)
Read More about the Properties of Limits and Formulas related to Limit.
How to Evaluate of Limits
Limits can be solved with different methods depending on the type of form it exhibits for x = a.
- Determinate Forms
- Indeterminate Forms
Determinate Forms
If at x = a, f(x) yields a definite value then the limit is calculated by \lim_{x \to a}f(x)=f(a).
Indeterminate Forms
If at x = a, f(x) yields a value in the form of 0/0, ∞/∞, ∞-∞, 00,1∞, and ∞0 then they are called Indeterminate Forms. It can be solved by following mentioned methods:
It is used when \lim_{x \to a}\frac{f(x)}{g(x)} takes the form of 0/0 then x-a is a factor of the numerator and denominator which can be cancelled to make it into determinate form and then solve.
This method is used when \lim_{x \to a}\frac{f(x)}{g(x)} takes the form of 0/0 or ∞/∞ and the denominator is in square root form. In this case, the denominator is rationalized.
In this case, the x in f(x) is replaced with x = a + h or a - h such that when x tends to a then h tends to 0.
When x→∞: In this case when \lim_{x \to ∞}\frac{f(x)}{g(x)} takes the form of ∞/∞ then the numerator and denominator are divided by the highest power of x.
Learn More, Strategy in Finding Limits
L Hospital Rule
L Hospital Rule states that if f(x)/g(x) is in the form of 0/0 or ∞/∞ for x = a then \lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)} , where f'(x) and g'(x) are the first order derivatives of functions f(x) and g(x) respectively.
Sandwich Theorem
Sandwich Theorem states that for given functions f(x), g(x), and h(x) that exists in the order f(x) ≤ g(x) ≤ h(x) for x belonging to the common domain then for some value 'a' if \bold{\lim_{x \to a}f(x)} = p = \bold{\lim_{x \to a}h(x)} then \bold{\lim_{x \to a}g(x)} = p
Continuity, Discontinuity, and Differentiability of a Function
The conditions for continuity, discontinuity, and differentiability of a function at a point are tabulated below:
| Continuity | Discontinuity | Differentiability |
|---|
| \lim_{x \to a^{-}}f(x)=\lim_{x \to a^{+}}f(x)=f(a) | - f(a) not defined
- Either Left Hand Limit or Right LImit doesn't exist or infinite
- \lim_{x \to a^{-}}f(x)\neq\lim_{x \to a^{+}}f(x)
- \lim_{x \to a^{-}}f(x)=\lim_{x \to a^{+}}f(x)\neq{f(a)}
| - \lim_{x \to a}\frac{f(x)-f(a)}{x-a} has finite value
- If Left Hand Derivative Right Hand Derivative i.e. \lim_{x \to a^{-}}\frac{f(x)-f(a)}{x-a}=\lim_{x \to a^{+}}\frac{f(x)-f(a)}{x-a}
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Mean Value Theorem
Mean Value Theorem states that if a function f(x) is continuous in the closed interval [a,b] and differentiable in the open interval (a,b) then there exists a point c in (a,b) such that
f'(c) = [f(b) - f(a)]/(b-a)
Learn More, Rolle’s Theorem and Lagrange’s Mean Value Theorem
Derivatives
Derivative is defined as the change in the output of a function with respect to the given input. This change is used to analyze the various physical factors associated with the function. Now we will look at the basic expression of Derivatives
Learn More, Algebra of Derivatives.
Differentiation Formulas
Some of the most common formula used to find derivative are tabulated below:
| d/dx(c) | 0 |
| d/dx{c.f(x)} | c.f'(x) |
| d/dx(x) | 1 |
| d/dx(xn) | nxn-1 |
| d/dx{f(g(x))} | f'(g(x)).g'(x) |
| d/dx(ax) | ax.ln(a) |
| d/dx{ln(x)} {Note: ln(x) = loge(x)} | 1/x, x>0 |
| d/dx(logax) | 1/xln(a) |
| d/dx(ex) | ex |
| d/dx{sin(x)} | cos(x) |
| d/dx{cos(x)} | -sin(x) |
| d/dx{tan(x)} | sec2x |
| d/dx{sec(x)} | sec(x).tan(x) |
| d/dx{cosec(x)} | -cosec(x).cot(x) |
| d/dx{cot(x)} | -cosec2(x) |
| d/dx{sin-1(x)} | 1/√(1 - x2) |
| d/dx{cos-1(x)} | -1/√(1 - x2) |
| d/dx{tan-1(x)} | 1/(1+x2) |
Read more about Differentiate the Function of a Function or Chain Rule.
Other Differentiation Techniques
Some other differentiation techniques includes:
Applications of Derivatives
Derivatives are used extensively in our daily lives, from calculating the speed of a moving vehicle to optimizing business decisions and understanding natural phenomena. In addition to real-life applications, derivatives are also used to solve various problems and help explain complex concepts. Some such use cases in mathematics are:
Differential Equation
Differential Equation refers to an equation that has a dependent variable, an independent variable, and a differential coefficient of the dependent variable with respect to the independent variable.
Order and Degree of Differential Equation
| Order of Differential Equation | Degree of Differential Equation |
|---|
Highest Derivative in the Equation Example: In (dy/dx)2 + 3(d2y/dx2)3 Order is 2. | Exponent raised to Highest Derivative Example: In (dy/dx)3 + 3(d2y/dx2)2 Degree is 3. |
Read more about Order and Degree of Differential Equation.
Types of Differential Equation
There are many types of differential equations based on various parameters, such as:
Also Read, Solution of Differential Equation.