Differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f . ∞ 1 If f(x) = (-1)"9"x" 1 + 9x n=0 ∞ f'(x) => n=1

2. Use differentiation to find a power series representation for f(x) = 1 (1 + x)²

Use Term by Term differentiation to find the power series for f(x)

1-Solve the D.E: * 1- Non exact (3x²y + 2xy + y)dx + (x2 + y?)dy = o e* (x*y +) = c ン3 u = e3x u = yー = C 3 1- None of them u = e2x 3 4

Let f(x) = cos (4x³) - 1 x5 Evaluate the 7th derivative of f at x = 0. f(7) (0) = 0 Hint: Build a Maclaurin series for f(x) from the series for cos(x).

Q5: a\. Let f(x) = x³ - 3x² - 1,x2 2. State and prove the theorem of derivative of inverse functions and use them to find the value of df-1/dx at the point x = -1 = f(3). b\ Write the Pascal's triangle for values of n = 6, then use it to determine the binomial series expansion of (3 - 2x)6, and simplify the result.

(2) Let f (x) bea function which is represented by the power series f(r) = (-1)" - 1)". n3 n=1 What is the fifth derivative of f at x 1?

Expand f(x) = In x in powers of (x-1) using Taylor's Series. Compute In 1.2. Step 1: Write the given function f (x) = Step 2: Get its derivative (at least until fourth derivative) f (x) = f"(x) = f"(x) = fV (x) = Step 3: Let a = 1 (From x-1, where a =1) f (1) = f' (1) = = (1) ייr fV (1) = Step 4: Recall Taylor's Series and substitute values of f(0), f (0), r'(0) .. f'(a)(x=a), f"(a)(x-a)² f"(a)(x=a)³ 1! f"(a)(x-a)" f(x) = f (a) + +...+ 2! 3! n ! f(x) = Step 5: Try to represent it in summation. f(x) = sin (x) = Eo() Step 6: Evaluate In 1.2. Using the series, let x= 1.2. Use your calculator for checking. f(1.2) = In (1.2) =