(1) Find the derivative of f(x) = x² + 2x using first principles as follows: ● Find f(x + h). • Calculate f(x + h) − f (x) and simplify it. • Simplify f(x+h)-f(x) • Now take the limit as h approaches zero, that is, limħ→ f(x+h)-f(x)¸

PLEASE FIND THE DERIVATIVE

2. a) Evaluate the following: -1 x-1(x+1)' xả-5x-20 x-5 x²-4x-5' x-5 i. lim ii. lim iii. lim- x-5x²-10x+25 √x²+4-2 x² 5h+4-2 iv. lim x-0 V. lim- h→0 h b) Find the derivative of the following functions: i. f(x) = -secx + 1 - secx ii. f(x) = x-5 tan 4x + sin 4x x iii. f(x) = ln(x-¹) [sinx + √x² + 1]. iv. f(t) = 3t² cost - t³ sin √t, e2x In(x²-3x+8) v. f(x) = (x+1)²

2.) Find the derivative Of 2x² using the limit definition of The derivative (Nd t

Find the derivative of the following: a.) f(x) = 5x6 +- 3x3/5 b.) f(x) = (x7 + 4x)(x5 – 2) (Hint: First expand.) 15/2 c.) f(x) = (47 –- 63) (Note: this is a constant function. It does not depend on x.)

If we need to calculate the derivative of a function at a point, there are two ways we can think about doing this. For example suppose f() = x – x, and we need to determine the value of f(4). One option is to just calculate the derivative at that point by plugging the point into the limit definition, like this: f(4 + h) – f(4) ( (4 + h)² – (4 + h)) - (4² – 4) f'(4) = lim (16 + 8h + h2 -4 – h) - (16 - 4) lim 7h + h2 = lim h(7+ h) lim h = lim h→0 h lim 7+h = 7. || h h h Another option is to calculate f'(x) in terms of x, and then plug in a value for x at the end, like this: f(x + h) – f(x) (2+ h) – (x + h)) - (2² – a) (22 + 2xh + h² – z – h) – (22 – 2) lim f'(x) = lim = lim 2xh + h2 - h h(2x + h - 1) = lim h 0 h h = lim = lim h h h that is to say, f'(x) = 2x – 1, and therefore f'(4) = 2(4) – 1 = 7. Notice that we get the answer f'(4) = 7 both ways. The second approach might look somewhat more complicated at first, but it turns out to be much more efficient if we ever need to know the…

Is the function f(2) = -2(xy+x)+i(x² – 2y – y²) differentiable? If it is, find it's derivative.

3. a) Use the Chain Rule and the fact that |x| = Vx2 to show that d %3D dx |x| b) Sketch the graphs of the function f(x) = |x| and its derivative. %3D c) Use the result of part (a) to differentiate the function g(x) = x|x|. Write the derivative as a simplified monomial. %3D