This problem finds the derivative of a function using the formal definition. The process is broken down into the following steps: Let f(x) be the function 10+6. Then the limit f(x+h)-f(x) h lim h-0 a b= C= d e= and 9 can be simplified to lim Evaluate the limit as h→ -a h 0 bx2 + cah+dz + eh+g to write the formula for f'(x) = 0 for:

Consider the function H(x) : 3 x + 2x 10x + 2 – 1 if x > 3 if x < 3 Determine if H is differentiable at x = 0 using the limit definition of the derivative.

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…

f(x + h) – f(x) Use the definition of derivatives using this formula lim h 2x -3 . 2. Find the derivative of f(x)= 3x +4