 In this video, we'll use a known Taylor series for a function to confirm the Taylor series for another function. And more globally, we'll talk about how operations with power series work. Consider the Maclaurin series for cosine of x. We have it written here. Now differentiating this term by term, we find that the derivative of the first term in this sum is zero. minus 2x over 2 factorial plus 4x cubed over 4 factorial minus 6x to the fifth over 6 factorial and so on. If we clean this up, we see this is equal to negative x plus x cubed over 3 factorial minus x to the fifth over 5 factorial and so on. Now if I factor out a negative sign, what I notice is that this is a familiar series to us. In fact, this is the infinite series for sine of x. So the derivative of cosine of x turns out to be negative sine of x. This confirms the claim in this section that the derivative of a power series is the sum of the derivatives of the terms in the series. Now integrating cosine of x, we have term by term we get x minus x cubed over 3 times 2 factorial which is 3 factorial plus x to the fifth divided by 5 times 4 factorial which is 5 factorial minus and so on. We recognize this again as the Maclaurin series for sine of x, of course, plus our constant c. So we again confirm that the integral of a power series is the sum of the integrals of the terms in the series as claimed in this section.