 Mathematics - Funktions 5
For functions with a bend, you can not create a clear tangent in the bend point. Since the derivation of a function is again a function which can be obtained, e.g. there is no derivation for the above function. If there is
one, then another derivative can be formed as f''(x).
Back to the rule of calculating a derivation. Perhaps you have noticed. Have a look at this result.
The exponent set as a multiplier before the x and in itself reduced by 1. Regardless of which constant (ie without x2 or x) is added, it disappears. Therefore, the following transpostions are vallid:
f(x) = x5 + 13
| f'(x) = 5 x4
|
As an additional function you have already learned about the sine or cosine. What is the first derivation of the sinus? We betray the result at first, namely
f(x) = sin x
| f'(x) = cosin x
|
The sinusoidal curve begins at f(x) = 0, that of the cosine at 1. The sinus has the slope 1 in the zero point. In the uppermost point of the sinusoidal curve, the slope is then zero again, which also accurately represents
the cosine curve. Also, at the point π with the slope -1.
The cosine thus gives exactly the slope of the sinusoidal curve for each x-value. Now there remains only the question after the derivation for the cosine. Here again these and more solutions:
f(x) = cosin x
| f'(x) = - sin x
|
f(x) = - sin x
| f'(x) = - cosin x
|
f(x) = - cosin x
| f'(x) = sin x
|
Or shortened here:
f(x) = sin x
|
f'(x) = cosin x
|
f''(x) = - sin x
|
f'''(x) = - cosin x
|
f(4)(x) = sin x
|
And back to one of our original examples:
f(x) = x3
|
f'(x) = 3 x2
|
f''(x) = 3 · 2x
|
f'''(x) = 6
|
f(4)(x) = 0
|
.
| 
