Differentiability.pdf - DIFFERENTIAL CALCULUS...

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1 D IFFERENTIAL CALCULUS D IFFERENTIABILITY LECTURE -04 1. Derivative: 2. Differentiability: 3. Differentiable Function: A function 𝑓(?) is said to be differentiable at ? = 𝑎 if 𝑓 (?) exists at ? = 𝑎 . That is 𝐿𝑓 (𝑎) = ?𝑓 (𝑎). 4. Left Hand Derivative (L. H. D.) and Right Hand Derivative (R. H. D.): If lim ℎ→0 𝑓(𝑎−ℎ)−𝑓(𝑎) −ℎ exists then it is called the left derivative of 𝑓(?) at ? = 𝑎 and denoted by 𝐿𝑓 (𝑎). Thus 𝐿𝑓 (𝑎) = lim ℎ→0 𝑓(𝑎−ℎ)−𝑓(𝑎) −ℎ . If lim ℎ→0 + 𝑓(𝑎+ℎ)−𝑓(𝑎) exists then it is called the right derivative of 𝑓(?) at ? = 𝑎 and denoted by ?𝑓′(𝑎). Thus ?𝑓 (𝑎) = lim ℎ→0 + 𝑓(𝑎+ℎ)−𝑓(𝑎) .
2 5. Geometrical Significance /Interpretation of differentiation:
3 Briefly, Let ? = 𝑓 (? ) be a function, and let ? (𝑎 , 𝑓( 𝑎) ) and ? (𝑎 + ℎ , 𝑓( 𝑎 + ℎ) ) be two points on the graph of the function that are close to each other. This graph is shown below: Joining the points ? and ? with a straight line gives us the secant on the graph of the function. And in the ∆??? the gradient of the line is given by In limiting process, i.e. as ? approaches ? , becomes really small, almost close to zero. So 𝑚 = change in y

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