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!
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.
For any specific derivative, you can ask a computer to check your result, as several other answers suggest.
However, if you want to be self-sufficient in taking derivatives (for an exam or other work), I recommend lots of focused practice. Most calculus textbooks include answers to the odd-numbered problems in the back of the book, and if you search for "derivative worksheet" you'll find lots of problem lists online.
Work through a list of at least 20 problems, and check your answers-- if you get less than 80% or 90% right, you know you need more practice. Here's the most important part: Track down your mistakes. Watch out for them in the future, and be sure you understand the right way to go. Pay attention to simplifying your answers, too, because a lot of people make algebra mistakes after getting the calculus right.
The rules you have are the best way to take these derivatives, you just have to be able to use them accurately.
Many derivative problems can be done more than one way. One way to check your work is to try them both ways and see if you get the same thing.
For example: $y=\frac{1}{x^2}$
You can write this as $y=x^{-2}$ and use the power rule to get $y'=-2x^{-3}=\frac{-2}{x^3}$
Or you can use the quotient rule $y'=\frac{0*x^2-2*x}{{(x^2)}^2}=\frac{-2x}{x^4}=\frac{-2}{x^3}$
This is probably too simple of an example, but I hope you get the idea.
I have a TI-84, and it has a Numeric Derivative function. It can be found by pressing the MATH button, and it is the 8th function in the list.
nDeriv( will appear on the screen. You then can enter in the original function, comma, the variable in the original function (probably $x$ unless you have something else stored), comma, and the value for x you wish for the calculator to plug in to the derivative.
For instance, if I am finding the value of the derivative of $x^3-5x+9$ at $x=10$, my screen would show this before pressing enter:
nDeriv(X^3-5X+9,X,10)
You then plug in the same x value to your supposed derivative which you are checking and manually solve it.
If the values match, your derivative is correct!
You can check certain values, like the saddle points, extremal points and local minima/maxima by setting the first derivative equal to zero/deriving further and checking these derivatives too.
If you found them right, putting the values into the original function plus/minus some $\Delta x$ should make things clear.
