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L7 Differentiation Rules QUOTIENT RULE COMPLETE.notebook
1
November 27, 2017
Rules for DifferentiationFinding the Derivative of a Quotient of Two Functions
Rewrite the function f(x) = as a function in polynomial form. Then, find f'(x).
Just as Leibniz was the first to publish a proof of the Product Rule for Differentiation, IsaacNewton was the first to publish a proof of the Quotient Rule of Differentiation using thelimit definition of the derivative. Let's write this rule together in the box below.
Quotient Rule of Differentiation
To show that this rule works, let's apply this rule to the function f(x) =that we rewrote and differentiated as a polynomial above.
Find the equation of the tangent line drawn to the graph of g(x) = when x = 2.
2x3 ‑ 3x2 + 2x2
2x3 ‑ 3x2 + 2x2
2x ‑ 1x + 3
If h(x) = , then h'(x) = g(x)f(x) f'(x)g(x) ‑ f(x)g'(x)
(g(x))2
L7 Differentiation Rules QUOTIENT RULE COMPLETE.notebook
2
November 27, 2017
We will now use the quotient rule to derive the derivative formulas for the remainingtrigonometric functions. Rewrite each function in terms of sine and/or cosine anddifferentiate using the Quotient Rule.
f(θ) = tan θ f(θ) = cot θ
f(θ) = sec θ f(θ) = csc θ
Find the equation of the normal line drawn to the graph of f(θ) = when θ = π. 3θ cos θ
L7 Differentiation Rules QUOTIENT RULE COMPLETE.notebook
3
November 27, 2017
Find the derivative of each of the functions below by applying the quotient rule.
f(x) = g(x) =
f(θ) = f(x) =
Find the equation of the normal line drawn to the graph of f(θ) = when θ = π.
3θ cos θ
x2 ‑ 2xx + 2
tan x x + 2
sin θ 1‑cos θ
3‑ x + 5
1x
L7 Differentiation Rules QUOTIENT RULE COMPLETE.notebook
4
November 27, 2017
Show, using the quotient rule, that if f(x) = , then f'(x) =x2 + 3x + 2x2 ‑ 1
3 (x ‑ 1)2‑
Similar to the Product Rule, there is a very valuable lesson that we must learn when weare introduced to the quotient rule. Below, first factor and simplify the functionf(x) = . Then differentiate using the quotient rule.x2 + 3x + 2
x2 ‑ 1
By Monday, read through Section 2.4‑‑The Chain Rule