Can we draw the graph of the derivative/integral of a function by using the graph of the function only?

Question

Consider a function say $F(x) = x^2 + 5\sin x$ then we have it's derivative as $F'(x) = 2x + 5\cos x$ and thus we have the graph of $F'(x)$ quiet easily but can we plot a graph using only the graph of $F(x)$ only? Since derivative signifies the slope of a curve at any point does it help us trace it's graph taking only the help of graph of $F(x)$?

At first I was wondering about the graph of the derivative then I wondered similarly what about $\int F(x) \, dx$ since it's the anti-derivative can we get its graph just from the graph of $F(x)$?

Answer 1

Yep, you could easily reconstruct a graph of either integral or derivative of the function. You could apply a numerical integration or differentian methods to obtain these graphs.

Based on request to provide some examples. I will use a rather simple function $ f(x) = x^2$ Suppose you would like to draw a graph of its dirivative on the following interval $[10;100]$. Algorithm is the following:

1. Break the interval into several sub intervals with, say, $step=10$;
2. Use the numerical approximation of the derivative i.e. $f'(x_{i}) = \frac{f(x_i)-f(x_{i-1})}{x_{i}-x_{i-1}} $ $(1)$where $x_{i}-x_{i-1} = step = 10$. This formula simply says that the true value of the dirivative is being approximated by tha numerical value.
3. Apply the $(1)$ for each interval;

Same idea could be applied for an intergral.