5.Use the formal definition of the derivative (we also called it the gr radient function): f'(x) = lim h→0 (f(x+h)-f(x) )/h, to find the derivative of f(x)=1/(x+1) [3 marks]

Step 1: Write down the function and the definition of the derivative.

The function is f(x) = 1/(x+1). The definition of the derivative is f'(x) = lim h→0 (f(x+h)-f(x) )/h.

Step 2: Substitute the function into the definition of the derivative.

f'(x) = lim h→0 ((1/(x+h+1)) - (1/(x+1))) / h

Step 3: Simplify the expression in the limit.

This can be done by finding a common denominator for the fractions in the numerator.

= lim h→0 ((x+1 - (x+h+1)) / ((x+h+1)(x+1))) / h = lim h→0 ((1 - h) / ((x+h+1)(x+1))) / h = lim h→0 ((1 - h) / (h(x+h+1)(x+1)))

Step 4: Simplify further by cancelling out the h in the numerator and denominator.

= lim h→0 ((1/h - 1) / ((x+h+1)(x+1))) = lim h→0 ((1/h) / ((x+h+1)(x+1)) - 1 / ((x+h+1)(x+1)))

Step 5: Take the limit as h approaches 0.

As h approaches 0, 1/h becomes infinity and 1 / ((x+h+1)(x+1)) becomes 1 / ((x+1)^2). Therefore, the derivative is:

f'(x) = -1 / ((x+1)^2)