Differentiating (x^2)(sinx) Using the Product Rule

Firstly, what is the product rule? What does it actually say? Well, it tells us how to differentiate a function of the form uv - the product of the functions u and v. If y = uv, the product rule says:

dy/dx = (du/dx)v + u(dv/dx)

So you differentiate the first bit, leaving the second part alone - giving you (du/dx)v. Then, you differentiate the second bit, leaving the first alone, - giving you u(dv/dx). And then you just add the two results to get dy/dx.

Let's look at applying this to the example in the question, trying to differentiate this: y = (x^2)(sinx)

We can see that y is a product of two functions, x^2 and sinx. Using the process above, we differentiate the first part, x^2, and leave sinx alone. That gives us (2x)(sinx). Then, we differentiate the second bit, sinx, and leave x^2 alone, and that gives us (x^2)(cosx). Then we just add the two together: (2x)(sinx) + (x^2)(cosx). So from that calculation we've shown that

dy/dx = (2x)(sinx) + (x^2)(cosx)

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