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Taking a calculus class, you’ll surely be asked to find derivatives of functions. Sometimes it’s not that easy to do. If your function is not among common ones, you need to apply special rules to find its derivative: product rule, quotient rule, chain rule, etc. In this section we’re going to show you an example of using chain rule. Suppose we need to find the first derivative of the following function:

h(x)=(2x^2+3)^{10}

In fact, in this particular case, we can just open the braces using binomial theorem and obtain a polynomial, derivative of which is easily found due to table of derivatives. But not that fast, we have 10th power and opening the braces would be a troublesome business. So let’s try to use chain rule instead, you’ll appreciate simplification it suggests. (more…)

If you’re asked to find a derivative of some function, sometimes it’s not a trivial thing to do. If you’re lucky and the function is among common functions, you can write down the required derivative right away using table of derivatives if necessary. But what if given expression is more complicated? That’s what this section is about. Let’s discuss chain rule for derivatives or so called outside-inside rule. This rule is needed when we want to find derivative of non-common function which is in fact composition of two or more common functions. (more…)

Earlier we’ve talked about gamma and beta functions.You’re probably interested, what beta function is useful for? In this section we’ll show how to integrate certain functions in simpler way with the help of beta function. Suppose we need to evaluate the following integral:

\int_0^{3}{x^{1/2}(27-x^3)^{-1/2} dx}

At first glance we can say that there is not an easy expression to deal with under the integral sign. As you can see we have radicals and powers under the integral sign so it’s not an easy thing to integrate directly. Moreover, take a look at the expression in braces: as it’s in negative power it means this is a denominator; and at the upper limit this denominator turns into zero. In other words, we’re facing an improper integral. And that’s what Euler’s functions are about! But which one should  we take – gamma or beta?  (more…)