MTH405_Wk_7_Lect_2

# MTH405_Wk_7_Lect_2 - MTH405 Wk 7 Lect 2 Techniques of...

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Unformatted text preview: MTH405 Wk 7 Lect 2 Techniques of Dierentiation In the last lecture, we discussed the denition of the derivative given by f (x + h) − f (x) . h→0 h f 0 (x) = lim The formula provides the benchmark for dierentiation but ofcourse, it is too tedious to use this formula to nd derivatives of more complicated functions. Fortunately, the denition of the derivative can be simplied for most functions into some rules listed below. • Power Rule • Trigonometric Table • Logarithmic and Exponential Rules • Product Rule • Quotient Rule • Chain Rule We will discuss all these rules as part of the coverage. 1 Notations of Dierentiation Given a function y = f (x), when we nd its derivative, there are a few notations we can use. The derivative can be denoted with f 0 (x) = f 0 = dy d [f (x)] = y 0 = . dx dx • Note • If it is dy du , then it means derivative of y with respect to u. • If it is du dx , then it means derivative of u with respect to x. dy dx means the derivative of y with respect to x. Properties of Dierentiation Just like limits, we have certain properties of dierentiation which helps us when nding derivatives. Theorem. Let f (x) and g(x) be functions, and k is a constant.Then the following properties hold. 1. The derivative of a constant is zero. d [k] = 0 dx 2. The derviative of x is 1 d [x] = 1 dx 3. Any coecient k multiplied to f (x) can be taken outside the derivative. d d [k · f (x)] = k · [f (x)] dx dx 4. We can dierentiate functions separately which are added together. d d d [f (x) + g(x)] = [f (x)] + [g(x)] dx dx dx 5. We can dierentiate functions separately which are subtracted. d d d [f (x) − g(x)] = [f (x)] − [g(x)] dx dx dx 2 Power Rule The Power Rule helps us in nding derivatives of functions of the form y = xn . For any function of the form y = xn , its derivative is Theorem. given by d n [x ] = n · xn−1 . dx Example. Dierentiate the following functions. 1. f (x) = x. 2. f (x) = 2x. 3. f (x) = 3x + 5. 4. f (x) = 2x2 + 13x − 2. 5. y = 4x5 + 7x4 − 3x2 + 16x − 20. 6. y = 3x 2 + 4x 3 + 3. 7. y= 5 8. y= 1 x2 √ 7 . x. 3 Trigonometric Table The table below lists down the derivatives of important trigonometric functions, where x is in radians. Function Derivative dy dx y = sin x dy dx y = cos x y = sec x y = csc x y = cot x = − sin x dy dx y = tan x dy dx dy dx = cos x = sec2 x = sec x tan x = − csc x cot x dy dx 4 = − csc2 x Exponential and Logarithmic Rules The table below lists down the derivatives of important exponential and logarithmic functions. Function Derivative dy dx y = ex dy dx y = ax = ax ln a dy dx y = ln x dy dx y = loga x 5 = ex = = 1 x 1 x ln a...
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