Вопрос от Анонимного юзера 05 марта 2025 19:08

Question 1

1. Вычислите производную функции: (C)' = (x)' = (2x)' = (Cx)' = (x ^ p) ^ ; = (x ^ 2)' = (1/x)' = (sqrt(x)) := (sin x)' = (cos x)' = (3x ^ 5) ^ ; = (x ^ 7)' = 2. Вычислите производную функции: (f(x) + g(x))' = (3x ^ 7 * sqrt(x)) "= (f(x) * g(x))' = ((f(x))/(g(x))) := (1/x + 3x ^ 7 + 12)' = (f(g * (x))) "= ((x ^ 4)/(sin x)) := (sqrt(cos x)) "= ((2x ^ 5 + 10) ^ 2)' = (sqrt(3x - 2)) ^ , 3. Вычислите производную в точке: 1) y = 2x ^ 3 - x ^ 2 + 3 x_{0} = 2 2) y = x ^ 2 * (x ^ 2 - 3) x_{0} = - 1 3) y = (x ^ 3 - 1)/(x + 2) x = 1 4) y = (2x - 1) ^ 5 - 7x, x_{0} = 1 5) y = - 3/x - 3x, x_{0} = - 1 6) y = (x + 1) ^ 5 - 1, 5x ^ 2 x_{0} = - 1 7) y = 1/((x - 2) ^ 7) x_{0} = 1 8) y = 1.5x ^ 2 x_{0} = 1 9) y = sqrt(x) , x_{0} = 4

Answer

(C)' = 0; (x)' = 1; (2x)' = 2; (Cx)' = C; (x^p)' = p*x^(p-1); (x^2)' = 2x; (1/x)' = -1/x^2; (sqrt(x))' = 1/(2sqrt(x)); (sin x)' = cos x; (cos x)' = -sin x; (3x^5)' = 15x^4; (x^7)' = 7x^6.
(f(x) + g(x))' = f'(x) + g'(x); (3x^7 * sqrt(x))' = 21x^6 * sqrt(x) + 3x^7 * (1/(2sqrt(x))); (f(x) * g(x))' = f'(x)g(x) + f(x)g'(x); ((f(x))/(g(x)))' = (f'(x)g(x) - f(x)g'(x))/(g(x))^2; (1/x + 3x^7 + 12)' = -1/x^2 + 21x^6; (f(g(x)))' = f'(g(x))g'(x); ((x^4)/(sin x))' = (4x^3 * sin x - x^4 * cos x)/(sin x)^2; (sqrt(cos x))' = -1/(2sqrt(cos x)) * sin x; ((2x^5 + 10)^2)' = 4(2x^5 + 10)(10x^4); (sqrt(3x - 2))' = 3/(2sqrt(3x - 2)).
1. y' = 12; 2) y' = -2; 3) y' = 3/9; 4) y' = 10; 5) y' = 3; 6) y' = -40; 7) y' = -7; 8) y' = 3; 9) y' = 1/4.

Question 2

1. Вычислите производную функции: (C)' = (x)' = (2x)' = (Cx)' = (x ^ p) ^ ; = (x ^ 2)' = (1/x)' = (sqrt(x)) := (sin x)' = (cos x)' = (3x ^ 5) ^ ; = (x ^ 7)' = 2. Вычислите производную функции: (f(x) + g(x))' = (3x ^ 7 * sqrt(x)) "= (f(x) * g(x))' = ((f(x))/(g(x))) := (1/x + 3x ^ 7 + 12)' = (f(g * (x))) "= ((x ^ 4)/(sin x)) := (sqrt(cos x)) "= ((2x ^ 5 + 10) ^ 2)' = (sqrt(3x - 2)) ^ , 3. Вычислите производную в точке: 1) y = 2x ^ 3 - x ^ 2 + 3 x_{0} = 2 2) y = x ^ 2 * (x ^ 2 - 3) x_{0} = - 1 3) y = (x ^ 3 - 1)/(x + 2) x = 1 4) y = (2x - 1) ^ 5 - 7x, x_{0} = 1 5) y = - 3/x - 3x, x_{0} = - 1 6) y = (x + 1) ^ 5 - 1, 5x ^ 2 x_{0} = - 1 7) y = 1/((x - 2) ^ 7) x_{0} = 1 8) y = 1.5x ^ 2 x_{0} = 1 9) y = sqrt(x) , x_{0} = 4

Answer

1. (C)' = 0; (x)' = 1; (2x)' = 2; (Cx)' = C; (x^p)' = p*x^(p-1); (x^2)' = 2x; (1/x)' = -1/x^2; (sqrt(x))' = 1/(2sqrt(x)); (sin x)' = cos x; (cos x)' = -sin x; (3x^5)' = 15x^4; (x^7)' = 7x^6. 2. (f(x) + g(x))' = f'(x) + g'(x); (3x^7 * sqrt(x))' = 21x^6 * sqrt(x) + 3x^7 * (1/(2sqrt(x))); (f(x) * g(x))' = f'(x)g(x) + f(x)g'(x); ((f(x))/(g(x)))' = (f'(x)g(x) - f(x)g'(x))/(g(x))^2; (1/x + 3x^7 + 12)' = -1/x^2 + 21x^6; (f(g(x)))' = f'(g(x))g'(x); ((x^4)/(sin x))' = (4x^3 * sin x - x^4 * cos x)/(sin x)^2; (sqrt(cos x))' = -1/(2sqrt(cos x)) * sin x; ((2x^5 + 10)^2)' = 4(2x^5 + 10)(10x^4); (sqrt(3x - 2))' = 3/(2sqrt(3x - 2)). 3. 1) y' = 12; 2) y' = -2; 3) y' = 3/9; 4) y' = 10; 5) y' = 3; 6) y' = -40; 7) y' = -7; 8) y' = 3; 9) y' = 1/4.

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