This is a summary of concepts under the topic of derivatives. It contains theorems, definitions and some examples. This will help you in your last-minute preparation for assessments.
Derivatives
Definition of Derivative
𝑓(𝑎+ℎ)−𝑓(𝑎)
• Gradient of PQ = represents average rate of
ℎ
change of 𝑓(𝑥) with respect to 𝑥 over the interval [𝑎, 𝑎 + ℎ]
if ℎ > 0.
𝑓(𝑎+ℎ)−𝑓(𝑎) 𝑓(𝑥)−𝑓(𝑎)
• 𝑙𝑖𝑚 = 𝑙𝑖𝑚 = 𝑓 ′ (𝑎)
𝑛→0 ℎ 𝑥→𝑎 𝑥−𝑎
• If 𝑓′(𝑎) exists, 𝑓 is differentiable at 𝑎
• If 𝑓′(𝑎) exists, 𝑓 ′ (𝑎)is also the instantaneous rate of change
of 𝑓(𝑥) with respect to 𝑥 at point 𝑎.
Instantaneous vs Average rate of change of 𝑓
2
𝑦 = 𝑥2
Let this 𝑦 = 𝑥 graph represent a velocity-time graph and let
𝑓(𝑥) = 𝑥 2 1.
1. Average rate of change of velocity from 𝑡 = 0 to 𝑡 = 2 is
𝑓(2)−𝑓(0) 22 −02
= = 2 𝑚/𝑠 2 2.
2−0 2−0
′ (1)
2. Instantaneous rate of change of velocity at 𝑡 = 1 is 𝑓 =
𝑓(1+ℎ)−𝑓(1) 2
𝑙𝑖𝑚 = 2 𝑚/𝑠
𝑛→0 ℎ
Estimation of Derivative
value of an instantaneous rate of change at a point by calculating the average rate of change over an appropriate small interval
𝑓(𝑎+ℎ0 )−𝑓(𝑎)
including that point: ≈ 𝑓 ′ (𝑎). This is true of 𝑓 ′ (𝑥) exists and ℎ0 is sufficiently small and nonzero. Such approximation
ℎ0
is essentially finding the gradient of the chord joining 2 points that surrounds point 𝑎 closely.
Differentiation from First Principles
Let 𝑓(𝑥) = √𝑥, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ [0, ∞). Find the value of 𝑓 ′ (4) from first principles.
𝑓(𝑥) − 𝑓(4) √𝑥 − √4 √𝑥 − 2 1 1 1
𝑓 ′ (4) = 𝑙𝑖𝑚 = 𝑙𝑖𝑚 = 𝑙𝑖𝑚 = 𝑙𝑖𝑚 = =
𝑥→4 𝑥−4 𝑥→4 𝑥 − 4 𝑥→4 (√𝑥 − 2)(√𝑥 + 2) 𝑥→4 √𝑥 + 2 √4 + 2 4
Relationship Between Differentiability and Continuity
Theorem: If 𝑓 is differentiable at 𝑎, then 𝑓 is continuous at 𝑎. The inverse holds true as well.
Chain Rule (Newton’s Notation)
Let 𝑓(𝑥) = sin 𝑥 𝑎𝑛𝑑 𝑔(𝑥) = 2𝑥. Find (𝑓 ∘ 𝑔)′(𝜋).
(𝑓 ∘ 𝑔)′ (𝜋) = 𝑓 ′ (𝑔(𝜋)) ∗ 𝑔′ (𝜋) = 𝑓 ′ (2𝜋) ∗ 2 = (cos 2𝜋) ∗ 2 = 2
Finding Limits using Derivatives
𝑓(𝑥)−𝑓(𝑎)
Suppose we wish to find 𝑙𝑖𝑚 𝑔(𝑥), where 𝑔(𝑥) can be written in the form for some function 𝑓. Then 𝑙𝑖𝑚 𝑔(𝑥) = 𝑓 ′ (𝑎).
𝑥→𝑎 𝑥−𝑎 𝑥→𝑎
tan(𝑥)
Evaluate 𝑙𝑖𝑚 .
𝑥→0 𝑥
tan(𝑥) tan(𝑥) − tan (0) 𝑑
𝑙𝑖𝑚 = 𝑙𝑖𝑚 = tan(𝑥)|𝑥=0 = sec 2 0 = 1
𝑥→0 𝑥 𝑥→0 𝑥−0 𝑑𝑥
Differentiability of Inverse Functions
If 𝑓 is a differentiable one-one function on an open interval 𝑙, then 𝑓 −1 is also differentiable at any point 𝑏 in the range of 𝑓 at
1
which 𝑓’[𝑓 −1 (𝑏)] ≠ 0, and its derivative is (𝑓 −1 )1 (𝑏) = .
𝑓 ′[𝑓 −1 (𝑏)]
Suppose that a differentiable one-one function 𝑓 has a tangent line 𝑦 = 6𝑥 + 9 at the points (-2, -3). Evaluate (𝑓 −1 )′(−3).
1 1 1
(𝑓 −1 )′ (−3) = = =
𝑓 ′ (𝑓 −1 (−3)) 𝑓 ′ (−2) 6
Rolle’s Theorem
Let f be a function that satisfies all of the following:
1. Continuous on the closed bounded interval [𝑎, 𝑏]; 2. Differentiable on the open interval (𝑎, 𝑏); 3. 𝑓(𝑎) = 𝑓(𝑏)
Then there is at least one point 𝑐 ∈ (𝑎, 𝑏) such that 𝑓’(𝑐) = 0.
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