Question

Найдите производную функций: 1)f(x)=2x^7 2) f(x)=1/10x^-10 3) f(x)=-8*1/x 4) f(x)=11корень из x 5) f(x)=141 6) f(x)=x^13 7) f(x)=ctg x-3cosx+sin x 8) f(x)=(6x+5)^14

Answer 1

В первых примерах надо знать только производую степенной функции: (ax^n)' = an x^(n - 1):
$(2x^7)'=14x^6\\\left(\dfrac{x^{-10}}{10}\right)'=-x^{-11}\\(-8/x)'=8/x^2\\(11\sqrt x)'=\dfrac{11}{2\sqrt x}\\141'=0\\(x^{13})'=13x^{12}$

7) Появились еще и (ctg x)' = -1/sin^2 x, (cos x)' = -sin x, (sin x)' = cos x:
$\left(\mathrm{\mathop{ctg}} x-3\cos x+\sin x\right)'=-\dfrac1{\sin^2x}+3\sin x+\cos x$

8) Производная функции от линейного аргумента: $(f(ax+b))'=af'(ax+b)$
$((6x+5)^{14})'=6\cdot14(6x+5)^{13}=84(6x+5)^{13}$

Answer 2

1) f(x) = 2x^7

To find the derivative of the function f(x) = 2x^7, we can use the power rule of differentiation. According to the power rule, if we have a function of the form f(x) = ax^n, where a and n are constants, then the derivative of f(x) with respect to x is given by:

f'(x) = n * a * x^(n-1)

Applying the power rule to the given function, we have:

f'(x) = 7 * 2 * x^(7-1) = 14x^6

Therefore, the derivative of f(x) = 2x^7 is f'(x) = 14x^6.

2) f(x) = 1/10x^-10

To find the derivative of the function f(x) = 1/10x^-10, we can apply the power rule of differentiation. The power rule states that if we have a function of the form f(x) = ax^n, where a and n are constants, then the derivative of f(x) with respect to x is given by:

f'(x) = n * a * x^(n-1)

Applying the power rule to the given function, we have:

f'(x) = -10 * (1/10) * x^(-10-1) = -x^-11 = -1/x^11

Therefore, the derivative of f(x) = 1/10x^-10 is f'(x) = -1/x^11.

3) f(x) = -8*1/x

To find the derivative of the function f(x) = -8*1/x, we can use the quotient rule of differentiation. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then the derivative of f(x) with respect to x is given by:

f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2

In this case, g(x) = -8 and h(x) = 1/x. Let's find the derivatives of g(x) and h(x) first:

g'(x) = 0 (derivative of a constant is 0) h'(x) = d/dx (1/x)

To find the derivative of 1/x, we can rewrite it as x^(-1) and apply the power rule:

h'(x) = -1 * x^(-1-1) = -1 * x^(-2) = -1/x^2

Now, we can apply the quotient rule:

f'(x) = (0 * 1/x - (-8) * (-1/x^2)) / (1/x)^2 = (8/x^2) / (1/x)^2 = (8/x^2) * (x^2/1) = 8

Therefore, the derivative of f(x) = -8*1/x is f'(x) = 8.

4) f(x) = √x

To find the derivative of the function f(x) = √x, we can use the power rule of differentiation. The power rule states that if we have a function of the form f(x) = x^n, where n is a constant, then the derivative of f(x) with respect to x is given by:

f'(x) = n * x^(n-1)

In this case, n = 1/2, since the square root can be written as x^(1/2). Let's apply the power rule:

f'(x) = (1/2) * x^((1/2)-1) = (1/2) * x^(-1/2) = 1/(2√x)

Therefore, the derivative of f(x) = √x is f'(x) = 1/(2√x).

5) f(x) = 141

The function f(x) = 141 is a constant function, and the derivative of a constant is always zero. Therefore, the derivative of f(x) = 141 is f'(x) = 0.

6) f(x) = x^13

To find the derivative of the function f(x) = x^13, we can use the power rule of differentiation. The power rule states that if we have a function of the form f(x) = x^n, where n is a constant, then the derivative of f(x) with respect to x is given by:

f'(x) = n * x^(n-1)

Applying the power rule to the given function, we have:

f'(x) = 13 * x^(13-1) = 13x^12

Therefore, the derivative of f(x) = x^13 is f'(x) = 13x^12.

7) f(x) = cot(x) - 3cos(x) + sin(x)

To find the derivative of the function f(x) = cot(x) - 3cos(x) + sin(x), we need to apply the chain rule and the derivative rules for trigonometric functions.

The derivative of cot(x) with respect to x is -csc^2(x) (the negative cosecant squared of x). The derivative of cos(x) with respect to x is -sin(x) (the negative sine of x). The derivative of sin(x) with respect to x is cos(x) (the cosine of x).

Applying these rules, we have:

f'(x) = -csc^2(x) - 3(-sin(x)) + cos(x) = -csc^2(x) + 3sin(x) + cos(x)

Therefore, the derivative of f(x) = cot(x) - 3cos(x) + sin(x) is f'(x) = -csc^2(x) + 3sin(x) + cos(x).

8) f(x) = (6x + 5)^14

To find the derivative of the function f(x) = (6x + 5)^14, we can use the chain rule and the power rule of differentiation.

Let's define g(x) = 6x + 5 and h(x) = x^14. Applying the chain rule, we have:

f'(x) = h'(g(x)) * g'(x)

To find g'(x), we can differentiate g(x) = 6x + 5, which is a linear function, and the derivative of a linear function is its slope:

g'(x) = 6

To find h'(x), we can apply the power rule of differentiation:

h'(x) = 14 * x^(14-1) = 14 * x^13

Now, let's substitute g'(x) and h'(x) into the chain rule formula:

f'(x) = h'(g(x)) * g'(x) = 14 * (6x + 5)^(14-1) * 6 = 14 * (6x + 5)^13 * 6

Therefore, the derivative of f(x) = (6x + 5)^14 is f'(x) = 14 * (6x + 5)^13 * 6.