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 Dierentiation In the last lecture, we discussed the denition of the derivative given by f (x + h) − f (x) . h→0 h f 0 (x) = lim The formula provides the benchmark for dierentiation but ofcourse, it is too tedious to use this formula to nd derivatives of more complicated functions. Fortunately, the denition of the derivative can be simplied 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 Dierentiation 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 Dierentiation Just like limits, we have certain properties of dierentiation 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 coecient k multiplied to f (x) can be taken outside the derivative. d d [k · f (x)] = k · [f (x)] dx dx 4. We can dierentiate functions separately which are added together. d d d [f (x) + g(x)] = [f (x)] + [g(x)] dx dx dx 5. We can dierentiate 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. Dierentiate 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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