Use the limit definition to compute the derivative of the function f(x)= 102/101-x at x=−51, and find an equation of the tangent line. f′(−51) =___. At x=−51, the tangent line is y =___.

Answer 1

The limit definition of the derivative is the following notation:

f'(x) = lim _h→0 [f(x+h) - f(x)] ÷ h

Step 1: Find f(x+h) = 102/(101-(x+h)) → 102/(101-x-h)

Step 2: Substitute f(x+h) and f(x) into the limit derivative equation as follows:

f'(x) = lim _h→0 [ (102/101-x-h) - (102/101-x)] ÷ h

Step 3: Clear and simplify the fraction using algebra/arithmetic by deriving the common demoninator (CD) as (101-x)(101-x-h) and apply distribution throughout the fraction to get:

Step 4: f'(x) = lim _h→0 [102(101-x) - (102(101-x-h)) ÷ (CD)} ×1/h

Step 5: Simplify numerator as follows:

f'(x) = lim _h→0 [102(101)-102x - (102(101) -102x-102h) ÷ (CD)} ×1/h

f'(x) = lim _h→0 [102(101)-102x - (102)(101) +102x +102h) ÷ (CD)} ×1/h

Step 6: Clear or cancelled out opposite pairs/terms, yielding:

f'(x) = lim _h→0 [102h ÷ (CD)] ×1/h

Step 7: Replace CD with its algebraic expression to get:

f'(x) = lim _h→0 [102h ÷ (101-x)(101-x-h)] ×1/h

Step 8: The h in the numerator and the 1/h divide each other out, yielding:

f'(x) = lim _h→0 [102 ÷ (101-x)(101-x-h)]

Step 9: Plug in 0 for the remaining h to evaluate the limit, hence derive the derivative as:

f"(x) = 102 ÷ (101-x)²

***NOTE*** Later on you will learn the quotient rule/technique to derive derivatives of fractions quicker than the limit definition, thought you can think of limit definition as the "formal" way to derive the first derivative, the equation that generates the slope at any given point along the original function's curve/path.