I can calculate the derivative of a function using the product rule, chain rule or quotient rule.
When I find the resulting derivative function however, I have no way to check if my answer is correct!
How can I check if the calculated derivative equation is correct? (ie I haven't made a mistake factorising, or with one of the rules).
I have a graphics calculator.
Thanks!
Best Answer
You can confirm it numerically by approximating its value at some particular points. By definition: $$f\,'(x)=\lim_{\Delta x \to 0}{\frac{f(x + \Delta x) - f(x)}{\Delta x}}$$
Example: $$\sin(x)'=\cos(x)$$ Let $\Delta x=0.000001$ and $x=1$ $$\sin(1)'\approx\frac{\sin(1+0.000001)-\sin(1)}{0.000001}\approx\frac{0.841471525 - 0.841470985}{0.000001}\approx0.54030189$$
Now your derivative: $$\cos(1)\approx0.54030231$$
Which is very close. The smaller $\Delta x$, the closer the result. Pick a number of points on x-axis and check them like that.
You can probably define a custom function using your calculator and build its graph: $$ \mathrm{error}(x)=\left|\frac{f(x+0.0000001)-f(x-0.0000001)}{0.0000002}-f\,'(x)\right|$$
Take into account that the approximation error needs to be compared to the value of your derivative at that point.