Questions and Answers #81 Derivatives

Questions and Answers Sheet 81 Derivatives Question #1 Find the derivatives of the functions 𝑞 = sin ( Answer: Given: The given function 𝑞 = sin ( 𝑡 √𝑡+1 ). find the derivative: Let 𝑓 = sin(𝑢) , = 𝑢 = 𝑡 √𝑡+1 𝑑𝑓(𝑢) 𝑑 𝑑 𝑡 [sin(𝑢)] [ = ] 𝑑𝑡 𝑑𝑢 𝑑𝑡 √𝑡 + 1 1 ) √𝑡 + 1(1) − 𝑡 ( 2√ 𝑡 + 1 = cos(𝑢) [ ] 2 (√𝑡 + 1) = cos(𝑢) [ = cos(𝑢) [ 2(𝑡 + 1) − 𝑡 2√𝑡 + 1(𝑡 + 1) 𝑡+2 2√𝑡 + 1(𝑡 + 1) ] ] 𝑡 cos ( ) (𝑡 + 2) 𝑡+1 √ = 2√𝑡 + 1(𝑡 + 1) Question #2 Find the derivatives of the following sin(𝑎𝑥 + 𝑏) cos(𝑐𝑥 + 𝑑) Answer: Step 1 Given sin(𝑎𝑥 + 𝑏) cos(𝑐𝑥 + 𝑑) Step 2 differentiating wrt x Let 𝑦 = sin(𝑎𝑥+𝑏) cos(𝑐𝑥+𝑑) Differentiating w.r.t x 𝑡 √𝑡+1 ) 𝑑 𝑑 𝑑𝑦 cos(𝑐𝑥 + 𝑑) 𝑑𝑥 sin(𝑎𝑥 + 𝑏) − sin(𝑎𝑥 + 𝑏) 𝑑𝑥 cos(𝑐𝑥 + 𝑑) 𝑑𝑥 | cos(𝑐𝑥 + 𝑑) |2 = cos(𝑐𝑥 + 𝑑) ∗ cos(𝑎𝑥 + 𝑏) 𝑎 − sin(𝑎𝑥 + 𝑏) (− sin(𝑐𝑥 + 𝑑) ∗ 𝑐) cos 2 (𝑐𝑥 + 𝑑) = 𝑎 cos(𝑎𝑥 + 𝑏) cos(𝑐𝑥 + 𝑑) + 𝑐 sin(𝑎𝑥 + 𝑏) sin(𝑐𝑥 + 𝑑) cos 2 (𝑐𝑥 + 𝑑) Question #3 Find the first and second derivatives 𝑦 = 𝑥2 + 𝑥 + 8 Answer: Step 1 The derivative of a function at a point is the rate of change of the function with respect to the function variable at that point which is equal to the slope of the function at that point . The second derivate of a function is the rate of change of the rate of change of the function at the given point w.r.t the function variable . Step 2 To determine the first derivative, of the function 𝑓(𝑥) = 𝑥 2 + 𝑥 + 8 , differentiate the function with respect to x as follows; 𝑓(𝑥) = 𝑥 2 + 𝑥 + 8 𝑓 ′ (𝑥) = 𝑑(𝑓(𝑥)) 𝑑𝑥 = 𝑑(𝑥 2 + 𝑥 + 8) 𝑑𝑥 = 𝑑(𝑥 2 ) 𝑑(𝑥) 𝑑(8) + + 𝑑𝑥 𝑑𝑥 𝑑𝑥 =2∗𝑥+1∗1+0 = 2𝑥 + 1 Therefore the first derivative of a function is 2𝑥 + 1 Step 3 Now to determine the second derivative, differentiate f'(x) with respect to x again, as follows; 𝑓 ′ (𝑥) = 2𝑥 + 1 𝑓 ′′ (𝑥) = 𝑑(𝑓 ′ (𝑥)) 𝑑𝑥 = 𝑑(2𝑥 + 1) 𝑑𝑥 = 𝑑(2𝑥) 𝑑(1) + 𝑑𝑥 𝑑𝑥 =2∗1+0 =2 Therefore the second derivative of the function is 2. Question #4 Find the first and second derivatives 𝑦 = 8𝑥 −7 − 6 𝑥 Answer: Step 1 Given 𝑦 = 8𝑥 −7 − 6 𝑥 To Find: The first and second derivative of the given function. Step 2 Solution: 𝑦 = 8𝑥 −7 − 6 𝑥 𝑦 = 8𝑥 −7 − 6𝑥 −1 Differentiating with respect to x we get 𝑑𝑦 𝑑𝑥 = 𝑑 𝑑𝑥 (8𝑥 −7 − 6𝑥 −1 ) 𝑑𝑦 𝑑 𝑑 = 8 (𝑥 −7 ) − 6 (𝑥 −1 ) 𝑑𝑥 𝑑𝑥 𝑑𝑥 We know that ⇒ 𝑑 𝑑𝑥 (𝑥 𝑛 ) = 𝑛𝑥 𝑛−1 𝑑𝑦 = 8(−7)𝑥 −8 − 6(−1)(𝑥 −2 ) 𝑑𝑥 Hence 𝑑𝑦 𝑑𝑥 = −56𝑥 −8 + 6𝑥 −2 Again differentiating with respect to x we get = −56 𝑑 −8 𝑑 (𝑥 ) + 6 (𝑥 −2 ) 𝑑𝑥 𝑑𝑥 = −56(−8)𝑥 −9 + 6(−2)𝑥 −3 = 448𝑥 −9 − 12𝑥 −3 Hence 𝑑2𝑦 𝑑𝑥 2 = 448𝑥 −9 − 12𝑥 −3 Question #5 Find the derivative of the following functions 𝑦 = ln(𝑥 4 + 1) Answer: Step 1 𝑦 = ln(𝑥 4 + 1) 𝑑2𝑦 𝑑𝑥 2 = 𝑑 𝑑𝑥 (−56𝑥 −8 + 6𝑥 −2 ) 𝑑𝑦 𝑑 = ln(𝑥 4 + 1) 𝑑𝑥 𝑑𝑥 Step 2 𝑑𝑦 1 𝑑 4 (𝑥 + 1) = 4 𝑑𝑥 𝑥 + 1 𝑑𝑥 𝑑𝑦 4𝑥 3 = 4 𝑑𝑥 𝑥 + 1 Question #6 Find the derivative of the following functions 𝑦 = ln(𝑒 𝑥 + 𝑒 −𝑥 ) Answer: Step 1 𝑦 = ln(𝑒 𝑥 + 𝑒 −𝑥 ) Take derivative both side, 𝑑𝑦 𝑑𝑥 = 𝑑 𝑑𝑥 ln(𝑒 𝑥 + 𝑒 −𝑥 ) Step 2 𝑑𝑦 1 𝑑 𝑥 (𝑒 + 𝑒 −𝑥 ) = 𝑥 −𝑥 𝑑𝑥 𝑒 + 𝑒 𝑑𝑥 𝑑𝑦 1 𝑑𝑒 𝑥 𝑑𝑒 −𝑥 = 𝑥 ( + ) −𝑥 𝑑𝑥 𝑒 + 𝑒 𝑑𝑥 𝑑𝑥 𝑑𝑦 1 (𝑒 𝑥 − 𝑒 −𝑥 ) = 𝑑𝑥 𝑒 𝑥 + 𝑒 −𝑥 𝑑𝑦 𝑒 𝑥 − 𝑒 −𝑥 = 𝑑𝑥 𝑒 𝑥 + 𝑒 −𝑥 Question #7 Find the derivative of the following functions 𝑦 = 10ln 2𝑥 Answer: Step 1 We use the chain rule 𝑦 = 10ln(2𝑥) 𝑑𝑦 𝑑𝑥 = 10ln(2𝑥) ln(10) 𝑑 𝑑𝑥 ln(2𝑥) Step 2 We again use the chain rule 𝑑𝑦 1 𝑑 (2𝑥) = 10ln(2𝑥) ln(10) 𝑑𝑥 2𝑥 𝑑𝑥 𝑑𝑦 1 = 10ln(2𝑥) ln(10) (2) 𝑑𝑥 2𝑥 𝑑𝑦 1 = 10ln(2𝑥) ln(10) 𝑑𝑥 𝑥 𝑑𝑦 10ln(2𝑥) ln(10) = 𝑑𝑥 𝑥 Answer: 10ln(2𝑥) ln(10) 𝑥 Question #8 Find the derivative of the following functions 𝑦 = ln √𝑥 4 + 𝑥 2 Answer: Step 1 𝑦 = ln √𝑥 4 + 𝑥 2 𝑑𝑦 𝑑 = (ln √𝑥 4 + 𝑥 2 ) 𝑑𝑥 𝑑𝑥 = 1 √𝑥 4 + 𝑥2 𝑑 (√𝑥 4 + 𝑥 2 ) 𝑑𝑥 Step 2 𝑑𝑦 1 (4𝑥 3 + 2𝑥) = 𝑑𝑥 2(𝑥 4 + 𝑥 2 ) 𝑑𝑦 4𝑥 3 + 2𝑥 = 𝑑𝑥 2(𝑥 4 + 𝑥 2 ) = 2𝑥 3 + 𝑥 𝑥4 + 𝑥2 = 2𝑥 2 + 1 𝑥3 + 𝑥 Question #9 Find the derivatives of the functions defined as 𝑦 = tan 𝑥 𝑥−1 Answer: Step 1 Consider the function: 𝑦 = tan 𝑥 𝑥−1 . Find the derivative of y. 𝑢 ′ Apply quotient rule of derivative that is ( ) = 𝑣 Therefore, 𝑦 ′ = (𝑥−1)(tan 𝑥)′ −(𝑥−1)′ tan 𝑥 (𝑥−1)2 Step 2 Use (tan 𝑥)′ = sec 2 𝑥 and (𝑎𝑥 𝑛 ) = 𝑎𝑛𝑥 𝑛−1 𝑣𝑢′ −𝑣 ′ 𝑢 𝑣2 . Therefore, 𝑦 ′ = (𝑥−1)(sec2 𝑥)−tan 𝑥 (𝑥−1)2 Question #10 1 Find the derivatives of the functions 𝑦 = (𝑥 2 + 1) (𝑥 + 5 + ) 𝑥 Answer: Step 1 1 Given 𝑦 = (𝑥 2 + 1) (𝑥 + 5 + ) 𝑥 Differentiating y with respect to x, 𝑑𝑦 𝑑𝑥 = 𝑑 𝑑𝑥 1 𝑑 𝑥 𝑑𝑥 (𝑥 2 + 1) (𝑥 + 5 + ) + 1 (𝑥 + 5 + ) (𝑥 2 + 1) 𝑥 1 1 = 2𝑥 (𝑥 + 5 + ) + (1 − 2 ) (𝑥 2 + 1) 𝑥 𝑥 = 2𝑥 2 + 10𝑥 + 2 + 𝑥 2 + 1 − 1 − 1 𝑥2 = 3𝑥 2 + 10𝑥 − 𝑥 −2 + 2 Step 2 Hence, the derivative of the given function is 𝑑𝑦 𝑑𝑥 = 3𝑥 2 = 10𝑥 − 𝑥 −2 + 2. Question #11 Find the derivatives of the functions 𝑟 = 12 0 − 4 03 + 1 04 Answer: Step 1 The rate of change in value of dependent variable with respect to the change in value of an independent variable is known as the derivative of the function. According to the quotient rule of differentiation, 𝑑 ( 𝑓(𝑥) )= 𝑑𝑥 𝑔(𝑥) 𝑓 ′(𝑥)𝑔(𝑥)−𝑓(𝑥)𝑔′ (𝑥) 2 (𝑔(𝑥)) Also, the differentiation of the expression of the form 𝑥 𝑛 is 𝑛𝑥 𝑛−1 . So, we need to use the quotient rule of differentiation to find the derivative of the function. Step 2 Using the quotient rule of differentiation, differentiate the function 𝑟 = 𝑑𝑟 𝑑 12 4 1 = ( − 3 + 4) 𝑑0 𝑑0 0 0 0 = 𝑑 12 𝑑 4 𝑑 1 ( )− ( 3) + ( ) 𝑑0 0 𝑑0 0 𝑑0 04 = 0 − 12 0 − 4 ∗ 302 0 − 403 − + 02 06 08 =− 12 12 4 + − 02 04 05 Therefore, the derivative of the function is 𝑑𝑟 𝑑0 =− 12 02 + 12 04 − 4 05 . 12 0 − 4 03 + 1 04 with respect to θ as follows: Question #12 Find the derivative of the following functions 𝑦 = (𝑥 2 + 1) ln 𝑥 Answer: Step 1 Given: 𝑦 = (𝑥 2 + 1) ln 𝑥 we know that 𝑑 𝑑𝑢 𝑑𝑣 (𝑢 ∗ 𝑣) = 𝑣 +𝑢 … (1) 𝑑𝑥 𝑑𝑥 𝑑𝑥 Step 2 so, by using equation(1) 𝑑𝑦 𝑑 [(𝑥 2 + 1) ln 𝑥] = 𝑑𝑥 𝑑𝑥 = (ln 𝑥) (∵ 𝑑 2 𝑑 (𝑥 + 1) + (𝑥 2 + 1) (ln 𝑥) 𝑑𝑥 𝑑𝑥 𝑑 𝑛 𝑑 1 (𝑥 ) = 𝑛𝑥 𝑛−1 , (ln 𝑥) = ) 𝑑𝑥 𝑑𝑥 𝑥 = (ln 𝑥)(2𝑥 + 0) + (𝑥 2 + 1) = 2𝑥 ln 𝑥 + 1 𝑥 𝑥2 + 1 𝑥 Hence, derivative of given function is 2𝑥 ln 𝑥 + 𝑥 2 +1 𝑥 . Question #13 Find the derivative of the following functions 𝑦 = 𝑥𝑒 Answer: Step 1 Given: 𝑦 = 𝑥 𝑒 we know that 𝑑 𝑑𝑥 (𝑥 𝑛 ) = 𝑛𝑥 𝑛−1 … (1) Step 2 so, by using equation(1) derivative of given expression will be 𝑑𝑦 𝑑𝑥 = = 𝑒𝑥 𝑒−1 hence, derivative of given expression is 𝑒𝑥 𝑒−1 . Question #14 1 2 Find the derivatives of the functions 𝑦 = (5 − 2𝑥)−3 + ( + 1) 8 𝑥 4 𝑑 𝑑𝑥 (𝑥 𝑒 ) Answer: Step 1 We need to find the derivative of the function, 4 1 2 𝑦 = (5 − 2𝑥)−3 + ( + 1) 8 𝑥 Step 2 Now differentiating the given function, = −3(5 − 2𝑥)−4 𝑑𝑦 𝑑𝑥 = 𝑑 𝑑𝑥 4 1 2 [(5 − 2𝑥)−3 + ( + 1) ] 8 𝑥 3 𝑑 1 2 𝑑 2 (5 − 2𝑥) + × 4 ( + 1) ( + 1) 𝑑𝑥 8 𝑥 𝑑𝑥 𝑥 = −3(5 − 2𝑥)−4 (0 − 2) + 3 1 2 −2 × 4 ( + 1) ( 2 + 0) 8 𝑥 𝑥 3 2 = 6(5 − 2𝑥)−4 − 𝑥 −2 ( + 1) 𝑥 Question #15 Find the derivatives of the following function. 𝑦 = 43∗𝑥 2 𝑦′ = Answer: Step 1 To find: The derivative of 𝑦 = 43∗𝑥 2 Formula used: Chain rule of derivative: ′ (𝑓(𝑔(𝑥))) = 𝑓 ′ (𝑔(𝑥)) ∗ 𝑔(𝑥) The derivative of 𝑎 𝑥 is 𝑎 𝑥 ∗ ln 𝑎. Power rule of derivative: ln 𝑎𝑏 = 𝑏 ∗ ln 𝑎 Step 2 Calculation: 2 The function 𝑦 = 43𝑥 can be written as, 𝑦 = 64𝑥 2 2 2 The derivative of 𝑦 = 64𝑥 can be obtained as, 𝑦 ′ = 64𝑥 ∗ ln 6 4 ∗ = 2 ∗ 𝑥 ∗ 64𝑥 2 ∗ ln 26 2 = 12𝑥 ∗ 64𝑥 ∗ ln 2 𝑑 𝑑𝑥 (𝑥 2 ) 2 Thus, the derivative of given function is 𝑦 ′ = 12𝑥 ∗ 64𝑥 ∗ ln 2. Question #16 Calculating derivatives Find 𝑑𝑦 > 𝑑𝑥 for the following functions. 𝑦= 1 − cos 𝑥 1 + cos 𝑥 Answer: Step 1 Use the identities cos 0 = 1 − 2 sin2 0 = 2 cos 2 0 − 1 and simplify the rational function. Step 2 𝑦= = 1 − (1 − 2 sin2 𝑥) 1 + (2 cos 2 𝑥 − 1) 2 sin2 𝑥 2 cos 2 𝑥 = tan2 𝑥 (tan 𝑥 = sin 𝑥 ) cos 𝑥 Step 3 Now differentiate the function 𝑦 = tan2 𝑥 with respect to x. 𝑑𝑦 = 2 tan 𝑥 sec 2 𝑥 𝑑𝑥 Step 4 Conclusion: 𝑑𝑦 = 2 tan 𝑥 sec 2 𝑥 𝑑𝑥 Question #17 Calculating derivatives Find 𝑑𝑦 > 𝑑𝑥 for the following functions. 𝑦= sin 𝑥 1 + cos 𝑥 Answer: Step 1 Given: 𝑦= sin 𝑥 1 + cos 𝑥 𝑥 The given function can be written as, 𝑦 = 𝑥 2 sin(2) cos(2) 𝑥 2 cos2 ( ) 2 (as sin 2 𝑥 = 2 sin 𝑥 cos 𝑥 , cos 2 𝑥 = 2 cos 2 𝑥 − 1 ) 𝑦 = tan 𝑥 2 Step 2 Differentiating y with respect to x, 𝑑𝑦 𝑑𝑥 = 𝑑 𝑑𝑥 𝑥 (tan ( )) 2 𝑥 𝑑 𝑥 = sec 2 ( ) × ( ) 2 𝑑𝑥 2 1 𝑥 = sec 2 ( ) 2 2 Hence, 𝑑𝑦 1 𝑥 2 2 = sec 2 ( ). 𝑑𝑥 Question #18 If f is a differentiable function, find an expression for the derivative of: 𝑦= 2𝑓(𝑥) 𝑒𝑥 Answer: Step 1 given that 𝑦= 2𝑓(𝑥) 𝑒𝑥 differentiatee on both side we apply division rulee here Step 2 𝑑𝑦 𝑒 𝑥 2𝑓 ′ (𝑥) − 2𝑓(𝑥)𝑒 𝑥 = (𝑒 𝑥 )2 𝑑𝑥 𝑑𝑦 2𝑒 𝑥 (𝑓 ′ (𝑥) − 𝑓(𝑥)) = (𝑒 𝑥 )2 𝑑𝑥 𝑑𝑦 𝑑𝑥 = 2(𝑓 ′(𝑥)−𝑓(𝑥)) 𝑒𝑥 answer Question #19 Find the derivatives of the functions 𝑧= 4 − 3𝑥 3𝑥 2 + 𝑥 Answer: Step 1 The derivative of a function is the rate of change of the function with respect to the function variable. The derivative of a function of the form ′ (𝑓(𝑥)/𝑔(𝑥)) = 𝑓(𝑥) 𝑔(𝑥) can be calculated using the quotient rule, which is given by, 𝑔(𝑥)∗𝑓 ′ (𝑥)−𝑓(𝑥)∗𝑔′ (𝑥) (𝑔(𝑥)) 2 Step 2 Therefore, the derivative of the function 𝑧 = 4−3𝑥 3𝑥 2 +𝑥 is given by, 𝑑( ′ 𝑧 = 4 − 3𝑥 ) 3𝑥 2 + 𝑥 𝑑𝑥 𝑑(4 − 3𝑥) 𝑑(3𝑥 2 + 𝑥) (3𝑥 2 + 𝑥) ∗ ( ) − (4 − 3𝑥) ∗ ( ) 𝑑𝑥 𝑑𝑥 = 2 2 (3𝑥 + 𝑥) = (3𝑥 2 + 𝑥) ∗ (−3) − (4 − 3𝑥) ∗ (6𝑥 + 1) (3𝑥 2 + 𝑥)2 = −9𝑥 2 − 3𝑥 − (24𝑥 + 4 − 18𝑥 2 − 3𝑥) (3𝑥 2 + 𝑥)2 = −9𝑥 2 − 3𝑥 − 24𝑥 − 4 + 18𝑥 2 + 3𝑥 (3𝑥 2 + 𝑥)2 = 9𝑥 2 − 24𝑥 − 4 (3𝑥 2 + 𝑥)2 Therefore, the derivative of the function is 9𝑥 2 −24𝑥−4 (3𝑥 2 +𝑥)2 Question #20 Find the derivatives of the functions 𝑦 = 9𝑥 2 +2 𝑥 2 +3 Answer: Step 1 Given: 𝑦= 9𝑥 2 + 2 𝑥2 + 3 To find: 𝑑 9𝑥 2 + 2 [ ] 𝑑𝑥 𝑥 2 + 3 Step 2 𝑑 9𝑥 2 + 2 [ ] 𝑑𝑥 𝑥 2 + 3 𝑑 𝑑 2 [9𝑥 2 + 2] ∗ (𝑥 2 + 3) − (9𝑥 2 + 2) ∗ [𝑥 + 3] 𝑑𝑥 𝑑𝑥 = 2 2 (𝑥 + 3) = (9 ∗ 2𝑥 + 0)(𝑥 2 + 3) − (9𝑥 2 + 2)(2𝑥 + 10) (𝑥 2 + 3)2 = 18𝑥(𝑥 2 + 3) − 2𝑥(9𝑥 2 + 2) (𝑥 2 + 3)2 = 18𝑥 2𝑥(9𝑥 2 + 2) − (𝑥 2 + 3) (𝑥 2 + 3)2