Question

How do you find the derivative of a power series?

Answer 1

Hint : As we know that derivative shows the rate of change that means the way a function changes at a certain point. For functions that have real numbers, it is a slope data appointed on the graph of a tangent line. We know that a power series is an infinite series of the form $\sum\limits_{n = 0}^\infty {{a_n}{{(x - c)}^n} = {a_0} + {a_1}{{(x - c)}^1} + ...}$

Complete step by step solution:
Let us assume a power series of the form $f(x) = \sum\limits_{n = 0}^\infty {{c_n}} {x^n}$ .
We know that one of the most useful properties of power series is that we can take the derivative term by term. Now by applying the Power Rule to each term we can write, $f(x) = \sum\limits_{n = 0}^\infty {{c_n}} {x^n} = \sum\limits_{n = 1}^\infty {n{c_n}} {x^{n - 1}}$ . We should note that when $n = 0$ , the term is zero.
Hence to find the derivative of the power series, we take the derivative term by term.

Note : We should note that the power series is quite big so it should be solved carefully step wise when required. When the function is centred at zero, then we can say that the power series is generated through the Maclaurin series. While expanding the series, we have to be careful while assigning the value of the power as they increase gradually.