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Introduction |
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The derivative, measures the rate at which the dependent variable changes with respect to the independent variable. It is one of the most important ideas in Calculus. The differentiation of functions are widely used in science, economics, medicine and computer science. |
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Sir Isaac Newton and Leibnitz share the credit of initiating development in the subject matter of derivability. |
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In this chapter, we learn the concept of derivability, process of differentiation, methods of differentiation and also obtain the second order derivatives of functions. |
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Derivative at a Point |
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Let f be a function and a be any point in its domain. Let h>0 be a small number. |
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f(x) is said to be differentiable if 
exists and is denoted by f|(a), then f|(a) is called the derivative or differential coefficient of f(x) at x = a. That is |
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There are two types of derivatives 1. Right Hand Derivative and 2. Left Hand Derivative. |
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Interpretation of Derivative at a Point |
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Physical Significance: Let Q (t) be a quantity that changes with time 't'. |
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Let Dt be the increment given to 't' and DQ be the corresponding increment in Q, then 
is called the average rate of change of Q with respect to 't'. (Also known as Newton - quotient of f at t). |
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Derivative of a Function (in general) |
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The derivative of the function f with respect to a variable x is the function f ' whose value at x is |
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 provided the limit exists. |
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Differentiability |
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We have already defined the derivative of a function f(x) at a particular point 'a' and derivative of f(x) in general for the variable x as f'(a) and f'(x) respectively. The restriction in both the cases is that 'the limit must exist'. If |
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does not exist, then we say that the function is not differentiable.
If the above limit exists, we say the function f(x) is differentiable. |
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Derivative of Some Important Functions |
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The Derivative of Some Important Functions are: 1. Derivative of a Constant, 2. Derivative of xn where n is any integer, 3. Derivative of a Constant of a Function, 4. Derivative of Exponential Function, 5. Derivative of the Logarithmic Function, 6. Derivative of sin x, cos x, tanx, cot x, secx, cosec x, 7. Derivative of Sum of Two Functions. |
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Product Rule for Differentiation |
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'Derivative of the product of two functions = |
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first function x derivative of second function + second function x derivative of first function' |
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Quotient Rule for Differentiation |
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The Quotient Rule for Differentiation is:
 |
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Derivative of a Function of a Function |
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The differentiation of function of a function is known as 'chain rule'. The chain rule is probably the most widely used differentiation rule in mathematics. |
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If y is a differentiable function of u and u is a differentiable function of x, then the derivative of y with respect to x is equal to the derivative of y with respect to u times the derivative of u with respect to x. |
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Derivative of Inverse Trignometric Functions |
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The Derivative of Inverse Trignometric Functions includes: 1. sin-1x,
2. cos-1x,
3. tan-1x,
4. cot-1x,
5. sec-1x,
6. cosec-1x.
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Derivative of Implicit Functions |
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Till now, the functions that we have discussed, are explicitly functions of x. We have defined y in terms of x. Suppose we have an equation f(x,y) = 0, which cannot be put in the form of y=f(x) to differentiate in the usual way, we can still differentiate the equation f(x,y) = 0. This function in which y cannot be expressed in terms of x, is called an implicit function. |
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The process of differentiating implicit function is called implicit differentiation. |
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Logarithmic Differentiation |
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When we want to differentiate a function of the form f(x) g(x), we use logarithmic differentiation. |
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Let y = f(x) g(x) |
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Taking log on both sides, we have logy = g(x) logf(x). |
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Differentiating with respect to x, we get, |
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 |
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Differentiation of Parametric Functions |
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If x=f(t) and y=g(t), where x and y are dependant on the independent variable t, then |
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 t is called the parameter. |
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Differentiation by Substitution |
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Differentiation of certain functions seem to be very difficult, but by suitably substituting the independent variable with some trigonometric function or other functions, they can be differentiated easily. |
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If f(x) involves inverse trigonometric functions of algebraic functions, the substitutions simplify the function f(x) to be differentiated.
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Derivatives of Second Order |
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The order of a differential equation is equal to the highest derivative in the equation. The single-quote indicates differentiation. So x' is a first derivative, while x'' is a second derivative. |
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x' = 1/x is first-order. |
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x'' = -x is second-order |
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x'' + 2 x' + x = 0 is second-order. |
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Summary |
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1. f'(a) exists at x = a iff Lf' (a) = Rf'(a). Where Rf '(a) is Right hand derivative and Lf '(a) is Left hand derivative. |
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2. Derivability implies continuity. |
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3. Derivative of a constant function is zero. |
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4. The derivative of a product is the derivative of the first times the second, plus the first times the derivative of the second. |
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5. The derivative of a quotient is the derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared. |
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6. Chain Rule :
The derivative of f(quantity) is the derivative of f, evaluated at that quantity, times the derivative of the quantity. |
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7. Logarithmic differentiation is the technique of first taking the (natural) logarithm of both sides of an equation and then finding dy/dx using implicit differentiation. Logarithmic differentiation is a useful alternative to the product and quotient rules when finding derivatives of particularly complicated expressions. |
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Conclusion |
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We have studied various techniques of differentiation. Also, we have studied the method of obtaining higher order derivatives of functions which is useful in maxima and minima problems.
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Welcome! 
