Mean Value Theorem

What is Differential Calculus?

Differential calculus is a branch of calculus that deals with the study of rates of change of functions and the behavior of these functions in response to infinitesimal changes in their independent variables....

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)....

Properties of Limits

If there are two functions f(x) and g(x) such that their limits [Tex]\lim_{x \to a}f(x)  [/Tex]and [Tex]\lim_{x \to a}g(x)  [/Tex]exist then following properties are followed...

Limit Formulas

Some of the common formulas for limits are:...

Evaluation of Limits

Limits can be solved with different methods depending on the type of form it exhibits for x = a....

Continuity, Discontinuity, and Differentiability of a Function

The conditions for continuity, discontinuity, and differentiability of a function at a point are tabulated below:...

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...

Algebra of Derivatives

In this section, we will learn how to find the derivative of two functions given in the form of an algebraic expression....

Differentiation Formulas

Some of the most common formula used to find derivative are tabulated below:...

Chain Rule

When we need to differentiate the function of a function, we apply the chain rule. In Chain Rule, we first differentiate the first function and then differentiate the second function and write their derivatives in product form. Some of the examples are mentioned below:...

Implicit Differentiation

Implicit Differentiation is used when a function is not defined explicitly in terms of only one independent variable. In this case, the function is given as g(x,y). It should be noted that here y is equal to f(x). Hence, the differentiation is done in the following manner:...

Higher Order Derivatives

Higher Order Derivatives refer to the derivative of derivative of a function. In this, we first differentiate a function and find its derivatives and then again differentiate the derivative obtained for the first time. If differentiation is done two times then it is called Second Order Derivative and if done for ‘n’ times it is called nth order derivative. For a function defined as y = f(x), its higher-order derivatives are given as follows:...

Error

Differential Error is a method used to calculate error in output for a change in input. To calculate the differential error, follow the below-mentioned steps...

Approximation

Approximation is used to find the approximate value of non-perfect square roots or cube roots. To find the approximate value, use the following steps:...

Critical Point

Critical Point is the point where the derivative of the function is either zero or not defined. C is the critical point of the function f(x) if...

Concave Up and Concave Down

The condition for concave up and down is tabulated below:...

Inflection Point

The point at which the concavity of a function changes is called the Inflection Point. The below-mentioned steps can be used to find the inflection point:...

Tangent and Normal

For a curve defined by function f(x) and let us assume there is a Point P(x1,y1) on it. Then,...

Increasing and Decreasing Function

Let f(x) is a function differentiable on (a,b) then the function is...

Maxima and Minima

The condition for maxima and minima is given below:...

Fermat’s Theorem

Fermat’s Theorem states that if a function is differentiable at its local extremum then its derivative at that point must be zero i.e., if x = a is the local extrema of f(x) then f'(a) = 0....

Extreme Value Theorem

If f(x) is continuous in the closed interval [a,b] then there exists c ≥ a for which f(c) is minimum and d ≤ b for which f(d) is minimum. In short, for a ≤ c and d ≤ b, f(c) is the absolute minimum in the closed interval [a,b] and f(d) is the absolute maximum in the closed interval [a,b]....

First Derivative Test

The condition for first derivative test or local maxima and local minima is given as...

Second Derivative Test

If x = a is a critical point of f(x) such that f'(a) = 0 then if...

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...

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....

Solution of First Order and First Degree Differential Equation

The solution of a first-order and first-degree differential equation can be found by different techniques depending upon the category they belong to....

Differential Calculus FAQs

What is Calculus?...