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DIFFERENTIABILITY, DIFFERENTIATION RULES AND FORMULAS



Differentiability of a function. A function is said to be differentiable at a point x = x0 if it has a derivative there. A function is said to be differentiable on an interval if it is differentiable at every point of the interval.


In general a function is differentiable on an interval if the function is smooth on that interval, meaning it has no abrupt changes in direction on the interval. If a function has a sharp change in direction at some point the derivative won’t exist at that point. The functions encountered in elementary calculus are in general differentiable, except possibly at certain isolated points on their intervals of definition.


Differentiation rules and formulas. In the following rules and formulas u and v are differentiable functions of x while a and c are constants.



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The derivative of a constant is zero.



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The derivative of a variable with respect to itself is one.



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The derivative of the sum of two functions is equal to the sum of their separate derivatives.



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The derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.



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The derivative of the product of a constant and a function is the constant times the derivative of the function.



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The derivative of the quotient of a function by a constant is the derivative of the function divided by the constant.



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The derivative of y with respect to x is the reciprocal of the derivative of x with respect to y.



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The derivative of the quotient of two functions is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.



9. Let y be a function of u and u be a function of x i.e. y = f(u) and u = g(x). Then


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            where n is any real number (positive, negative, fractional, etc).



The derivative of the n-th power of x is n times the (n-1)th power of x.



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            where n is any real number (positive, negative, fractional, etc).



The derivative of the n-th power of a function is n times the (n-1)th power of the function times the derivative of the function.




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where u and v are both variable


 


Trigonometric functions


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Inverse trigonometric functions


The inverse trigonometric functions are multi-valued. The principal branches are as follows.



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Exponential and logarithmic functions



 




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            where a is a constant.



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Hyperbolic functions



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Inverse hyperbolic functions



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Derivation of formulas


We will now show the derivation of some of the above formulas to illustrate the manner in which they are derived.


Formula 1


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Derivation


Let u and v be differentiable functions of x. If, starting at any fixed value, x increases by an amount Δx, u and v will change by corresponding amounts Δu and Δv, respectively. Then

 

1)        y = u + v

2)        y + Δy = u + Δu + v + Δv


Subtracting 1) from 2) gives

 

3)        Δy = Δu + Δv


Dividing by Δx gives


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Formula 2



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Derivation


Let u and v be differentiable functions of x. If, starting at any fixed value, x increases by an amount Δx, u and v will change by corresponding amounts Δu and Δv, respectively. Then

 

1)        y = uv

2)        y + Δy = (u + Δu)(v + Δv) = uv + uΔv + vΔu + ΔuΔv


Subtracting 1) from 2) gives

 

3)        Δy = uΔv + vΔu + ΔuΔv


Dividing by Δx gives


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Formula 3


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Derivation


Let u and v be differentiable functions of x. If, starting at any fixed value, x increases by an amount Δx, u and v will change by corresponding amounts Δu and Δv, respectively. Then


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Subtracting 1) from 2) gives


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Formula 4


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Derivation


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Formula 5


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Derivation


Let u be a differentiable function of x and y a differentiable function of u. If, starting at any fixed value, x increases by an amount Δx, u will change by a corresponding amount Δu and y by an amount Δy, respectively. Then


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Taking limits of both sides as Δx →0


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Formula 6


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Derivation


Let u be a differentiable function of x and

 

1)      y = un .


Then an increment Δx in x produces a change Δu in u and a corresponding change Δy in y giving

 

2)        y + Δy = (u + Δu)n


Expanding (u + Δu)n with the Binomial Theorem gives


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Subtracting 1) from 3) gives


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All terms after the first have the limit zero as Δx →0. Thus taking limits of both sides as Δx →0 gives


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Note. This proof applies only for positive integral values of n. However the formula is valid for all values of n.




Formula 7


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Derivation


Let u be a differentiable function of x. Then

 

1)        y = sin u

2)        y + Δy = sin (u + Δu) = sin u cos Δu + cos u sin Δu


Subtracting 1) from 2) gives

 

3)        Δy = sin u (cos Δu - 1) + cos u sin Δu



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Now

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provide radian measure is used. Why? Two important theorems:


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Thus


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Formula 8


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Derivation


To differentiate cosine u we note that


            cos u = sin (π/2 - u)


and use the formula for differentiating the sin of a function. Thus


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Formula 9


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Derivation


The formula for the derivative of tan u can be obtained by writing


            tan u = sin u / cos u


and differentiating the quotient.



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