5.3 Derivatives of a Few Fundamental Functions

5.3.1 Powers and Polynomials

Normally one finds the derivatives of the most important functions in tables or one has learned them by heart in school. They are however very easy to calculate, if one takes into account, that the limit Δx 0 takes place and that therefore all higher powers of Δx can be neglected.

We show this in detail for the example of the second power:

y(x) = x2 y(x + Δx) = (x + Δx)2 = x2 + 2xΔx + Δx2 y(x + Δx) - y(x) = 2xΔx + Δx2 y(x+Δx)-y(x) Δx = 2x + Δx y = limΔx0y(x+Δx)-y(x) Δx = limΔx0(2x + Δx) = 2x

This can now be easily extended to arbitrary powers, if one takes into account that the second term of the polynomial (x + Δx)n = xn + nxn-1Δx + axn-2Δx2 + bxn-3Δx3 + .. is nxn-1Δx

The coefficients of the further terms a,b,c, do not have to be given explicitly, since all these terms vanish in the the limit Δx 0, since they contain at least the factor Δx2:

y(x) = xn y(x + Δx) = (x + Δx)n = xn + nxn-1Δx + axn-2Δx2 + cxn-3Δx3.. y(x + Δx) - y(x) = nxn-1Δx + axn-2Δx2 + bxn-3Δx3 + ... y(x+Δx)-y(x) Δx = nxn-1 + Δx(axn-2 + .....) y = limΔx0y(x+Δx)-y(x) Δx = limΔx0(nxn-1 + Δx(....)) = nxn-1

This also yields the rule for higher derivatives of powers:

y(x) = xn,y = nxn-1,y = n(n - 1)xn-2,.... y(n) = n(n - 1)(n - 2).....(1) = const y(n+1) = 0

The derivative of a constant c. which has by definition the same value of all values of the independent variable , is zero.

The rules obtained above also apply if the exponents are negative or rational:

y = x-n = 1 xn y = -nx-n-1 = -nx-(n+1) = - n xn+1 y = x3 = x13 y = 1 3x13-1 = 1 3x-23 = 1 3x23

With this result it is also easy to see how the derivative of polynomials look; example:

y = 3x5 + 4x4 + 3x - 1 y = 15x4 + 16x3 + 3 y = 60x3 + 48x2;y = 180x2 + 96x;y(4) = 360x + 96;y(5) = 360;y(6) = 0

We have shown the formal differentiation of powers in so much detail, because this also allows to treat functions for which a series expansion containing powers is known.

5.3.2 Exponential Function

In analogy to the exponential series we can define the exponential function for a continuous domain of the variable x. Since its series expansion consists of powers we can obtain its derivative immediately via differentiating its individual terms according to the rule derived above.

e  =  2,71828.... y =  ex=limn(Sn =   m=0nxm m! )1 + x + x2 12 + x3 123 + x4 1234... y = 0 + 1 + 2x 12 + 3x2 123 + 4x3 1234... = 1 + x + x2 12 + x3 123 + ... y = y y = y = y

Thus the exponential function has the property, that its derivatives and the function are identical. The above derivation also shows, that the coefficient of the n-th term of the exponential sequence 1 n! is given by the reciprocal of the n-th derivative of its respective power:

y = xn n! ;y = n xn-1 n! = xn-1 (n - 1)!;y(n) = x0 0! = 1;y(n+1) = 0

Upon differentiation every term assumes the form of the previous term and the constant term vanishes. This property results in the exponential function and its derivative becoming identical.

5.3.3 Trigonometric Functions

In an analogous manner we can obtain the derivatives of the trigonometric function from their series expansions. We start with the representations, that we previously obtained from the complex exponential function:

y = sinx = x -x3 3! + x5 5! - .. = 0(-1)n x2n+1 (2n+1)! y = 1 -3x2 3! + 5x5 5! + ... = 1 -x2 2! + x4 4! - ..... = 0(-1)n x2n (2n)! = cosx y = cosx = 1 -x2 2! + x4 4! + .. = 0(-1)n x2n (2n)! y = -2x 2! + 4x3 4! - ...... = -(x -x3 3! + x5 5! ...) = - 0(-1)n x2n+1 (2n+1)! = -sinx

By taking into account the signs all further derivatives can be established:

y = sinx y = cosx;y = -sinx;y = -cosx;y = sinx y = cosx y = -sinx;y = -cosx;y = sinx;y = cosx

Using these results all functions, that can be described as series expansions in terms of trigonometric functions, can be easily differentiated. Those are in the main such functions, that describe periodic phenomena.

5.3.4 Rules for the Differentiation of combined Functions

Combined functions are easy to differentiate if one knows the derivatives of the functions that are combined. The following, immediately plausible rules apply.

multiplicative constantc y = c f x y = c fx additive composition  y = f(x) + g(x) y = f(x) + g(x) product rule y = f(x) g(x) y = f(x) g(x) + f(x) g(x) quotient rule y = f(x) g(x) y = f(x)g(x)-f(x)g(x) (g(x))2 chain rule  y = f(g(x)) y = f(g(x)) g(x) exampley = sin(x3 + x) y = cos(x3 + x) * (3x2 + 1)

5.3.5 Derivatives of further Fundamental Functions

To be able to differentiate all “prevalent” functions formally, one needs a collection of derivatives of additional Fundamental functions. We list these here without comment in the form of a table together with those obtained above. The derivatives of the hyperbolic functions at the end of the table are simply obtained from their definitions in terms of exponential functions.

y = xn y = nxn-1 y = ex y = ex;y = eax y = aeax;y = ax y = ax lna y = sinx y = cosx;y = cosx y = -sinx y = tanx y = 1 cos 2x;y = cotx y = - 1 sin 2x y = arcsinx y = 1 1-x2 ;y = arccosx y = - 1 1-x2 y = arctanx y = 1 1+x2 ;y = arccotx y = - 1 1+x2 y = lnx y = 1 x;y = a logx y = 1 x ln a y = sinh(x) = ex-e-x 2 y = ex+e-x 2 = cosh(x) y = cosh(x) = ex+e-x 2 y = ex-e-x 2 = sinh(x)

In contrast to the trigonometric functions sine(x) and cosine(x) the derivatives of the hyperbolic functions sinh(x) and cosh(x) show no additional sign change on differentiation.

For the inverse functions of the trigonometric functions we make use of the notations such as arccos(x) that are employed in mathematical texts; in Java code we use instead acos(x).