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# Tutorial_5 - Page 1 Example 1 Find the derivative of the...

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Page 1 Example 1. Find the derivative of the function f by employing the appropriate rules. (a) f ( x ) = 5 p 2 ¡ 3 x 3 . (b) f ( t ) = 2 3 t +1 ¡ t ¡ 4 : (c) f ( x ) = (log 2 x ) 3 ;x > 0 Solution: (a) f 0 ( x ) = 5(2 ¡ 3 x 3 ) 1 = 2 ; so k 0 ( x ) = 5 ¢ 1 2 (2 ¡ 3 x 3 ) ¡ 1 = 2 ¢ (2 ¡ 3 x 3 ) 0 = 5 2 (2 x ¡ 3 x 3 ) ¡ 1 = 2 ( ¡ 9 x 2 ) = ¡ 45 2 x 2 (2 ¡ 3 x 3 ) ¡ 1 = 2 ± = ¡ 45 x 2 2 p 2 ¡ 3 x 3 : (b) f 0 ( t ) = 2 3 t +1 ¢ ln2 ¢ (3 t + 1) 0 ¡ ( ¡ 4) t ¡ 5 = 2 3 t +1 (3ln 2) + 4 t ¡ 5 . (c) f 0 ( x ) = 3(log 2 x ) 2 ¢ 1 x ln2 = 3 x ln2 (log 2 x ) 2 : Example 2. Find the derivative of the function by using the appropriate rules of diﬁerenti- ation. (a) y = ± e x x + 1 6 (b) y = ( x + 1)ln x; x > 0 : Solution: (a) y 0 = 6 ± e x x + 1 5 ¢ ± e x x + 1 0 = 6 ± e x x + 1 5 ¢ ± e x ( x + 1) ¡ e x ( x + 1) 2 = 6 e 5 x ( x + 1) 5 ¢ xe x ( x + 1) 2 = = 6 xe 6 x ( x + 1) 7 : (b) y 0 = ( x + 1) 0 ln x + ( x + 1) (ln x ) 0 = ln x + x + 1 x = ln x + 1 + 1 x . Example 3. Use implicit diﬁerentiation to ﬂnd dy dx : 2 x 3 y 2 ¡ y = x + 1 : Solution: Rewrite the equation with y = y ( x ): 2 x 3 [ y ( x )] 2 ¡ y ( x ) = x + 1 : Diﬁerentiate both sides with respect to x : 2 ¢ 3 x 2 ¢ [ y ( x )] 2 + 2 x 3 ¢ 2 y ( x ) ¢ dy dx ¡ dy dx = 1 :

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Tutorial_5 - Page 1 Example 1 Find the derivative of the...

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