Lists of derivatives and integrals

For problems and applications of Calculus we will need derivatives and integrals of given functions, in practice obtained from formula tables and with differentiation rules. The method is very simple most engineers and scientists would already be familiar with the basic functions and they may be looked up in lists of basic calculus formulas.

This work is not difficult anyone who reads this particular entry might be able to apply it.

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It is also explained how to use the tables only using tables of basic formulas for elementary functions used for problems. Lists of functions are readily available in the literature, i.e. tables of basic differentiation and integration formulas.

Usually given as two separate lists however in fact you really need only one.

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Given a function F in the first column (list), you look can up the derivative F' in the second list. This is standard.

However since given the derivative f in the second column, you may then find corresponding integral of f, the antiderivative F in the first one.

Integration is in general more difficult.

The derivative rules, addition multiplication etc. and other properties enables one to apply to many other and more complicated functions. The most important I think is the chain rule.

Note

We want to explain what the indefinite integral  F(x) + C  is, as function of x

According to the fundamental theorem,  F(x)  is the integral of  f(x)  and thus the antiderivative of f , the definite integral  F(x) – F(a)  with fixed a and constant C but variable now x = b
 
F '(x)  =  f(x) + C with C a constant.

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As typical examples with a differentiation table and rules,

The product rule,

F(x) = y = cos x . sin x   //   F '(x) =  dy/dx = cos x . cosx – sin x . sinx

K(x) = y =  x^2 exp(x)   //   K '( x) =  dy/dx = 2x.exp(x)  + x^2.exp x 

and sum rule,

G(x) = F(x) + K(x)   //   G '(x) = cos x . cosx – sin x  sinx + 2x.exp(x)  + x^2.exp x 

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The chain rule is a bit tricky   [ M ( N (x) ) ] ' = M '[ N '(x) ]  or easier to remember,

dz/dx = dz/dy . dy/dx  for example with  z = y^2  and  y = e^x  i.e.  z = (e^x) ^2 

dz/dx = (2y). e^x =  2e^x . e^x = 2e^2x

Or another,  z = sin (y)  with  y = x^4  as  z = sin( x^4)

dz/dx = dz/dy . dy/dx  = cos(y) . 4x^3 = 4x^3 cos( x^4)

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Anti-derivitives, integration examples directly from the tables, the integrals are then as the formulas on the right side.

F '(x) = x^n   //   F(x) = 1/(n+1) .x^(n+1) + C

G '(x) =  ln(x)   //   G(x) = x ln(x) – x + C

K '(x) = cos x   //   K(x) = -sin x + C

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You should try some exercises!

Of course you may also use the tables to confirm. For more complicated functions we may use more advanced methods such as integration by parts.