the derivative. definition example (1) find the derivative of f(x) = 4 at any point x
TRANSCRIPT
The Derivative
Definition
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Example (1)Find the derivative of f(x) = 4 at any point x
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Example (2)Find the derivative of f(x) = 4x at any point x
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Example (3)Find the derivative of f(x) = x2 at any point x
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Example (5)Find the derivative of f(x) = x4 at any point x
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Example (6)Find the derivative of f(x) = 3x3 + 5x2 - 2x + 7 at any point
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QuestionsFind from the definition the derivative of each of the following functions:
1 .f(x) = √x
2 .f(x) = 1/x
3 .f(x) = 1/x2
4 .f(x) = 5 / (2x + 3)
Power Rule
Let:
f(x) = xn , where n is a real number other than zero
Then:
f'(x) = n xn-1
If f(x) = constant , then f'(x) = 0
Algebra of Derivatives
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The Chain RuleThe derivative of composite function
for the case f(x) = gn(x)
Let:f(x) = gn(x)Then:f' (x) = ngn-1(x) . g'(x)
Example:Let f(x) = (3x8 - 5x + 3 )20
Then f(x) = 20 (3x8 - 5x + 3 )19 (24x7 - 5)
Examples (1)
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Answers of Questions(1)
Find from the definition the derivative of
each of the following functions:
1 .f(x) = √x
2 .f(x) = 1/x
3 .f(x) = 1/x2
4 .f(x) = 5 / (2x + 3)
1
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Differentiability & Continuity1 .If a function is differentiable at a point, then it is continuous at that point.
Thus if a function is not differentiable at a point, then it cannot be continuous at that point.
But the converse is not true. A function can be continuous at a point without being differentiable at that point.
2 .A point of the graph at which the graph of the function has a vertical tangent or a sharp corner is a point where the function is not differentiable regardless of continuity
Examples(1) Sharp Corner
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This function (Graph it!) is continuous at the point x=2, since the limit and value of the function at that point are equal ( Show that!) but it is not differentiable at that point, since the right derivative
of f at x=2 is not equal to the left derivative a that point.
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When both right and left derivatives are +∞ or both are - ∞
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Limits Involving Trigonometric Functions
All trigonometric functions are continuous a each point of their domains, which is R for the sine & cosine functions(→The limit of sinx and cosx at any real number a are
sina and cosa respectively)
, R-{ π/2, - π/2, 3π/2, - 3π/2,…………} for the Tangent and the Secant functions
and R-{0, π, - π, 3π, - 3π ,…………} for the Cotangent and the Cosecant functions.
Important Identity
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Solution
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Solution
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Example (4)
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All members are continuous for all x other than 2. For a member to be continuous at x=2, we need the limit of the member function at x= 2 to exist and to equal the value at that point, which is 1/2c. Since the right limit at x=2 of any member is equal o 1/2c, it remains that its left limit at that point be equal to that value.
Solution
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