 In this video, I want to show you how to compute the derivative of the function f of x equals four over x and we're going to compute this by the definition. So it's important to remember what is the derivative. The derivative is the limit of the difference quotient. So take the limit as h goes to zero of f of x plus h minus f of x all over h. So we want to compute this for which we can't just plug in h equals zero because we'll end up with zero over zero which is indeterminate form. So we're going to have to try to simplify the difference quotient algebraically. So in the context of our function four over x, if we look at the expression f of x plus h, that means we place the x in the denominator of f of x with an x plus h, like so. Then we have to subtract from it f of x which is just four over x and then we have an h right here. So this is an example of a nested fraction. We have a fraction with little fractions inside of it. In order to clean this thing up a little bit, I want you to identify the denominators of the little baby fractions. Oh, they're so adorable. We're going to look for the least common multiple of all of their denominators, the least common denominator of these babies, in which case this is going to be x times x plus h. Don't bother multiplying that out. It'll be more advantageous to keep it factored. So in order to keep the fraction proportional, we need to multiply the top of the mother fraction and the denominator also by this x times x plus h. In the numerator, we're going to distribute it for which we're going to then get the limit. We're going to get four times x times x plus h. This sits above x plus h. Then for the second one, we're going to get four times x times x plus h over, well, x, just the x right there, sorry. And then the main denominator, you're going to get h times x times x plus h. When it comes to fractions, do not multiply out the denominator. Keep it factored. There's not going to be any benefit of distributing h to the x in this context. So leave it factored. Well, because we multiply the mother fraction by the least common denominator of the baby fractions, you'll notice that the x plus h cancels with the first one, so it's no longer a fraction. And then the x cancels with the second one, so it's no longer a fraction. So this difference quotient simplifies to be 4x minus 4 times x plus h all over h times x times x plus h. Again, keep denominators factored. Never multiply out a numerator. I want you to make that oath upon your children's children's children's that you'll never multiply out a denominator here. On the other hand, numerators, please do multiply them out. Distribute that negative 4. Make sure the negative sign is carried with you there. You're going to get 4x minus 4x minus 4h over hx x plus h as h goes to zero. You're going to notice that the 4x will now cancel out. You're left with just a negative 4h in the numerator. You still have hx x times h as we take the limit as h is going to zero. So because of all the work we've done in the numerator, you're now going to notice that the numerator is just negative 4 times h. The factor of h in the numerator cancels with the factor of h in the denominator. In which case, then we get the limit of negative 4 over x times x plus h as h goes to zero. So although there's still an h in the denominator, do notice that if we send h equal to zero, if we send h equal to zero, we're not going to divide by zero anymore. We're going to get negative 4 over x times x plus zero. The denominator is actually going to become just an x squared. And this is then the derivative of our original function 4 over x.