1. Compute f'(a) using the limit definition of f at x = a. 2. Compute dy/dx using the limit definition. 3. Compute the derivative.

Steps to Solve

1. Deriving using the limit definition

For each function, the derivative can be computed using the limit definition:

$$ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} $$

2. Function (a): ( f(x) = x^2 - x ) at ( a = 1 )

First, calculate ( f(1) ):

$$ f(1) = 1^2 - 1 = 0 $$

Now compute ( f(1+h) ):

$$ f(1+h) = (1+h)^2 - (1+h) = 1 + 2h + h^2 - 1 - h = h^2 + h $$

Now apply the limit:

$$ f'(1) = \lim_{h \to 0} \frac{(h^2 + h) - 0}{h} = \lim_{h \to 0} (h + 1) = 1 $$

3. Function (b): ( f(x) = 5 - 3x ) at ( a = 2 )

Calculate ( f(2) ):

$$ f(2) = 5 - 3(2) = -1 $$

Now compute ( f(2+h) ):

$$ f(2+h) = 5 - 3(2+h) = 5 - 6 - 3h = -1 - 3h $$

Applying the limit:

$$ f'(2) = \lim_{h \to 0} \frac{(-1 - 3h) - (-1)}{h} = \lim_{h \to 0} \frac{-3h}{h} = -3 $$

4. Function (c): ( f(x) = x^{-1} ) at ( a = 4 )

Calculate ( f(4) ):

$$ f(4) = \frac{1}{4} $$

Now compute ( f(4+h) ):

$$ f(4+h) = \frac{1}{4+h} $$

Applying the limit:

$$ f'(4) = \lim_{h \to 0} \frac{\frac{1}{4+h} - \frac{1}{4}}{h} $$

Finding a common denominator:

$$ = \lim_{h \to 0} \frac{4 - (4+h)}{4(4+h)h} = \lim_{h \to 0} \frac{-h}{4(4+h)h} = \lim_{h \to 0} \frac{-1}{4(4+h)} = -\frac{1}{16} $$

5. Function (d): ( f(x) = x^3 ) at ( a = -2 )

Calculate ( f(-2) ):

$$ f(-2) = (-2)^3 = -8 $$

Now compute ( f(-2+h) ):

$$ f(-2+h) = (-2+h)^3 = -8 + 12h - 6h^2 + h^3 $$

Applying the limit:

$$ f'(-2) = \lim_{h \to 0} \frac{(-8 + 12h - 6h^2 + h^3) - (-8)}{h} = \lim_{h \to 0} \frac{12h - 6h^2 + h^3}{h} = \lim_{h \to 0} (12 - 6h + h^2) = 12 $$