Derivatives

posted on: 18 Jan, 2012 | updated on: 06 Sep, 2012

The Derivatives are the measure of the function that how a function changes as function’s input changes. In short derivatives Calculus describes the behavior of one function in accordance with the other function. In calculus derivatives explains the linear approximation of the function at a selected input near about that input value.
Let’s have a function y = f(x) that is a continuous function of x. Now suppose if there is an increment of δx is done in x then a small increment of δy will also be there in y, as y is a function of x. When δy/ δx tends to zero, then it is known as differential coefficient of y with respect to x. It is represented by dy/dx and the process of finding this differential coefficient is known as differentiation. The root method to find the derivatives is first principle. There are some standard derivatives below which are much useful in finding the derivatives of Functions.
For example:
1.       D(xⁿ)/dx = n* x^(n-1)
2.       D(a^x)/dx = a^x* log_ea
3.       D(sinx)/dx = cos x
4.       D(tanx)/dx = sec²x
5.       D(cotx)/dx = -cosec²x
6.       D(secx)/dx = secx * tanx
7.       D(cosecx)/dx = - cosecx cotx
8.       D(sin^-1x)/dx = 1/(√1-x²)
9.       D(cos^-1x)/dx = -1/(√1-x²)
10.    D(tan^-1x)/dx = 1/(1+x²)
11.    D(cot^-1x)/dx = -1/(1+x²)
12.    D(sec^-1x)/dx = 1/x√(x²-1)
13.     D(cosec^-1x)/dx = -1/x√(x²-1)
These are some standard derivatives which help to solve the problems and there are some fundamental theorems also which are used.
·         Derivative of a constant quantity is zero.
·         If there are more than one function to be Differentiated then they are separated first then any other operation is performed.
·         Derivative of the product of two Functions: it is calculated by the following formula.
           D(u*v)/dx = u*dv/dx + v* du/dx.

Topics Covered in Derivatives

Implicit Differentiation

posted on: 08 Feb, 2012 | updated on: 06 Sep, 2012

Before proceeding to the implicit differentiation we first understand, what is the exact meaning of implicit Functions and how we solve the Calculus implicit differentiation? Implicit function is a relation between two or more variable which cannot be solve. For example, in the equation x³ + y³= 5. In this example y is an implicit function of x.

Let’s ha...Read More

Linear Approximation in Calculus

posted on: 08 Feb, 2012 | updated on: 24 May, 2012

Linear approximation helps us to estimate the difficult to calculate type curved graph. It uses the values on the line which we want; this Point is very close to the graph. That is gives the almost desired values of the point. Linear approximation helps in many numerical techniques such as Newton Raphson Method to approximate the values. Let’s move on...Read More

Newton-Raphson Method

posted on: 08 Feb, 2012 | updated on: 24 May, 2012

Newton Raphson method is used for solving polynomial equations of the form g (x) = 0. Firstly, we make our own initial guess for the roots we are going to find and we use this initial guess i.e. x₀.

The sequence of values x₁, x₂, x₃, x₄ . . . are generated in the way, which is described below and should be nearer to the exact root. We need a formula to impl...Read More

Vertical Tangents and Cusps

posted on: 08 Feb, 2012 | updated on: 24 May, 2012

In mathematics vertical Tangent is a vertical line which has infinite Slope. A function whose graph has a vertical tangent cannot be differentiated at the Point of tangency.
A function ‘g’ would have a vertical tangent at ‘x = c’, if the difference quotient used to define the derivative which infinite limit, that is-
lim_h→0 g(a + h) – g(a) / h = + ∞,
or
...Read More

Rolle's Theorem

posted on: 08 Feb, 2012 | updated on: 02 Sep, 2012

According to Rolle's theorem if y = f(x) is continuous at every Point of the close interval [a,b] and differentiable at every point of its interior (a,b) and f(a) = f(b) then there is at least one number ‘c’ exist between a and b such that f¹(c) = 0.

According to definition of Rolle’s Theorem we can say that:

f¹(c) = f(b) – f(a) = 0.
...Read More

Monotonicity and the sign of Derivatives

posted on: 08 Feb, 2012 | updated on: 24 May, 2012

To understand the concept of monotonicity and the sign of Derivatives, let’s assume that we will investigate the function of "nice". Now we split the nice word into sub intervals and it is necessary that the function is monotone and differentiable as its Domain. But many cases not every function has this property. In other words if function do...Read More

Critical Points

posted on: 08 Feb, 2012 | updated on: 24 May, 2012

Critical points can be defined in many ways. Let’s understand it with help of followings steps:

(I) On a graph, where derivative has no value or zero or does not exist, is very important and has to be considering for solving the many application problems of the Derivatives. These specific points where this occurs actually are known as critical points.

Global Extrema

posted on: 08 Feb, 2012 | updated on: 24 May, 2012

In mathematics we deal with two type of extrema's i.e. global extrema and local extrema. Local extrema is a collection of local maxima and local minima, but global extrema is a collection of maximum value of local maxima and minimum value of local minima of Functions .For calculating maximum and minimum value of Functions we have to follow these steps-

Step 1...Read More

Concavity and Points of inflection

posted on: 08 Feb, 2012 | updated on: 24 May, 2012

Second derivative of a function is a measure tool to investigate the graph of a function because the turning Point of a function cannot be determined with the help of only Tangent line. It’s needed to study Concavity and Points of inflection.

Inflection point on a curve is a point where the curve changes its direction of curvature or concavity. ...Read More