Notes On Differentiability of a function - CBSE Class 12 Maths
The derivative of a real function f at point c in its domain is defined by: $\underset{h\to \text{0}}{\text{lim}}\frac{\text{f(c+h)-f(c)}}{\text{h}}$ Derivative of f(x) at c = f '(c) or d/dx[f(x)]c f ' (x) = $\underset{h\to \text{0}}{\text{lim}}\frac{\text{f(x+h)-f(x)}}{\text{h}}$ provided the limit exists. Let y = f(x) The derivative of y is denoted by y ' or dy/dx. Differentiation is the process of finding the derivative of a function If the limit does not exist, it means that function f is not differentiable at c. A function f is said to be differentiable at point c in its domain, if: $\underset{h\to {\text{0}}^{\text{-}}}{\text{lim}}\frac{\text{f(c + h) - f(c)}}{\text{h}}$   = $\underset{h\to {\text{0}}^{\text{+}}}{\text{lim}}\frac{\text{f(c + h) - f(c)}}{\text{h}}$   A function f is said to be differentiable in interval [a,b], if it is differentiable at every point of [a,b]. f ' (a) is called the right hand derivative and f ' (b) is called the left hand derivative of f. A function is said to be differentiable in (a,b), if it is differentiable at every point of (a,b). Theorem: If function f is differentiable at point c, then it is also continuous at that point. Proof: Given function f is differentiable at point c. By definition, f ' (c) = $\underset{x\to \text{c}}{\text{lim}}\frac{\text{f(x) - f(c)}}{\text{x - c}}$ For x ≠ c, we have f(x) - f(c) = (f(x) - f(c))/(x - c) x (x - c) = $\underset{x\to c}{\text{lim}}\text{[}\frac{\text{f(x) - f(c)}}{\text{x - c}}\text{}×\text{(x - c)]}$ ⇒ = $\text{}$ $\underset{x\to c}{\text{lim}}\text{(x - c)}$ = f ' (c) = 0 ⇒ = = f(c) Hence, f is continuous at x = c. Note: All differentiable functions are continuous, but the converse is not always true.

Summary

The derivative of a real function f at point c in its domain is defined by: $\underset{h\to \text{0}}{\text{lim}}\frac{\text{f(c+h)-f(c)}}{\text{h}}$ Derivative of f(x) at c = f '(c) or d/dx[f(x)]c f ' (x) = $\underset{h\to \text{0}}{\text{lim}}\frac{\text{f(x+h)-f(x)}}{\text{h}}$ provided the limit exists. Let y = f(x) The derivative of y is denoted by y ' or dy/dx. Differentiation is the process of finding the derivative of a function If the limit does not exist, it means that function f is not differentiable at c. A function f is said to be differentiable at point c in its domain, if: $\underset{h\to {\text{0}}^{\text{-}}}{\text{lim}}\frac{\text{f(c + h) - f(c)}}{\text{h}}$   = $\underset{h\to {\text{0}}^{\text{+}}}{\text{lim}}\frac{\text{f(c + h) - f(c)}}{\text{h}}$   A function f is said to be differentiable in interval [a,b], if it is differentiable at every point of [a,b]. f ' (a) is called the right hand derivative and f ' (b) is called the left hand derivative of f. A function is said to be differentiable in (a,b), if it is differentiable at every point of (a,b). Theorem: If function f is differentiable at point c, then it is also continuous at that point. Proof: Given function f is differentiable at point c. By definition, f ' (c) = $\underset{x\to \text{c}}{\text{lim}}\frac{\text{f(x) - f(c)}}{\text{x - c}}$ For x ≠ c, we have f(x) - f(c) = (f(x) - f(c))/(x - c) x (x - c) = $\underset{x\to c}{\text{lim}}\text{[}\frac{\text{f(x) - f(c)}}{\text{x - c}}\text{}×\text{(x - c)]}$ ⇒ = $\text{}$ $\underset{x\to c}{\text{lim}}\text{(x - c)}$ = f ' (c) = 0 ⇒ = = f(c) Hence, f is continuous at x = c. Note: All differentiable functions are continuous, but the converse is not always true.

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