# Thread: function of a derivative

1. ## function of a derivative

find a function whose derivative is 20x^3 plus 6x^2 plus 5

2. Originally Posted by eduk8tedidiot
find a function whose derivative is 20x^3 plus 6x^2 plus 5

What you want to find here is the 'anti-derivative' of the function. The integral of a derivative gives you the antiderivative.

Hence, if we are seeking function $f(x)$, then

$f(x) = \int 20x^3 + 6x^2 + 5 \, dx$

3. Originally Posted by eduk8tedidiot
find a function whose derivative is 20x^3 plus 6x^2 plus 5

Have you heard of an integral? Well anyway, we're looking for the anti-derivative of $20x^3+6x+62+5$

In order to get an $x^3$ the power rule tells us we need an $x^4$ but the derivative of $x^4=4x^3$ so we'd have a 4 out in front when we want a 20, so let's put a 5 out in front of our original guess

Now the derivative of $5x^4=20x^3$ so the first term is done....

Doing similar steps will lead you to an answer of $20x^4+2x^3+5x+c$

Note that when we take the derivative of that, the +c has a derivative of zero, since the question is to find an antiderivative, you can choose any value for c as the answer

4. $f'(x) = 20x^3 + 6x^2 + 5$

$f(x) = 5x^4 + 2x^3 + 5x + C$

$C = konst$

$f(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$

$f'(x) = a_1 + 2a_2x + 3a_3x^2 + ...+ na_nx^{n-1}$

$f(x) = C$ C = const. $\rightarrow f'(x) = 0$

$f(x) = kx^m \rightarrow f'(x) = m*k*x^{m-1}$

5. Originally Posted by josipive
$f'(x) = 20x^3 + 6x^2 + 5$

$f(x) = 5x^4 + 2x^3 + 5x + C$

$C = konst$

$f(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$

$f'(x) = a_1 + 2a_2x + 3a_3x^2 + ...+ na_nx^{n-1}$

$f(x) = C$ C = const. $\rightarrow f'(x) = 0$

$f(x) = kx^m \rightarrow f'(x) = m*k*x^{m-1}$