f x h f x dy f x y lim dx
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Aug 18, 2023 •289 likes •462 views

* Definition of Derivative: y f x The first derivative of the function with f respect to the variable x is the function whose value at x is: f x h f x dy

  3. * Definition of Derivative:    y f x The first derivative of the function with f  respect to the variable x is the function whose value at x is:       f x h f x dy        f x y lim  dx h h 0 provided the limit exists.

  4. Important theorems: 1. If y = f(x) is differentiable at x = a , then y = f(x) is continuous at a . The inverse is not always true. 2. If the function y = f(x) is discontinuous at the point x = a , then it is not differentiable at this point.

  5. Geometric Interpretation of Derivative: * The slope of tangent line to the graph of the function f(x) at (a,f(a)) is the derivative of f(x) at x = a . y   y  f x      . .   f a h f a P       h h x a  a h       f a h f a     f a lim  h h 0 = Slope of tangent at P

  6. * we can write the equation of the tangent line to the curve at the point (a, f(a)) :           y f a f a x a Example Find an equation of the tangent line to the curve    2 y x 8 x 9 at the point (3,- 6). Solution     y         3 2 3 8 2 y 2 x 8            y 2 x y 6 2 x 3

  7. Rules of Differentiation:     d d      c f x c f x dx dx   d     2 3 x 3 2 x 6 x dx         f g f g   d     3 2 x 6 x 5 3 x 6 dx

  8.        f g f g g f    d        3 2 3 2 x x x   dx              2 3 2 x 3 3 x 1 x x 2 2       f f g g f      2   g g             4 3  x x 1 x 1 4 x 1    d x 1        4 2 dx x x  4 x x

  9. Tables of Differentiation      f x f x Table (1) k (const ant) 0 k k x  1 n n nx x 1 x 2 x x x e e x x a a ln a 1 ln x x

  10. Table (2) Table (3) Trigonometric Functions Hyperbolic Functions           f x f x f x f x cos x sinx sinhx cosh x - cos x sin x cosh x sinh x 2 tan x 2 sec x tanh x sec h x - 2 - cot x 2 csc x csc h x coth x sec x - sec x tan x sec hx sec h x t nh a x - - csc cot x x csc x csc hx csc h x c th o x

  11. Example Differentiate the functions:   3 a y ) x sin x    2 y 3 x cos x  b y ) x cosh x 1      y x sinh x cosh x 2 x 3  x cos x  ) c y sin x      3  1 2 / 3 2 /     2 x sin x  sin x cos x x cos x     y 2 sin x

  12. Example: Obtain the derivative of tan x from sin x and cos x. Solution:   d   d sin x  tan x     dx dx cos x     cos cos sin sin x x x x  2 cos x  2 2 1 cos x sin x   2  sec x 2 2 cos x cos x

  13. The Derivative of a Composite Function     d         f g h x f g h x    dx     g h x     h x x  

  14. Examples Differentiate the functions:     y   2  2 cos x a ) y s n i x    2 x      3 ' 3   y sin x sin x ) y cos  si n  b x x   1   3 2 sin x x     2 3 x cos x

  15. Examples 1     1       tan sec hx     1 y tan cos 5 5   h x  1   4 5 /     tan sec y hx   5   2  sec sec hx  sec hx tanh x

  16. Example       3  x 2 2 ln co t y e x     3 2  x e ln cot x   3   1 2 x   x  ]   csc  [ 2 cot x y e   2 cot x   3   ] 2 2 [  x   3 x ln cot x e

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