Question

Найти поизводную. 1)y=-3корня из x+1/3cosx-1/2ctgx= 2)корень из x(-2x+1)= 3)Y=x/x^2-1= ЕСЛИ можно подробно

Answer 1

1)$y=-3 \sqrt{x} + \frac{1}{3cosx}- \frac{1}{2ctgx} =-3x^{ \frac{1}{2} } + \frac{1}{3cosx}- \frac{1}{2} tgx \\ \\ y'= -3*\frac{1}{2}*x^{ \frac{1}{2}-1 }+ \frac{1}{3} * \frac{sinx}{cos^2x} - \frac{1}{2} * \frac{1}{cos^2x}= -\frac{3}{2 \sqrt{x} } + \frac{2sinx-3}{6cos^2x}$

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2)$y= \sqrt{x(-2x+1)} = \sqrt{-2x^2+x}=(x-2x^2)^{ \frac{1}{2} } \\ \\ y'= \frac{1}{2} (x-2x^2)^{ -\frac{1}{2} }*(1-4x)= \frac{1-4x}{2 \sqrt{x-2x^2} }$

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3)$y= \frac{x}{x^2-1} \\ \\ y'= \frac{1*(x^2-1)-2x*x}{(x^2-1)^2} = \frac{x^2-1-2x^2}{(x^2-1)^2} =- \frac{x^2+1}{(x^2-1)^2}$

Answer 2

Finding Derivatives

To find the derivatives of the given functions, let's solve each one step by step.

1) Function: y = -3√x + 1/3cos(x) - 1/2ctg(x)

To find the derivative of this function, we can use the sum rule and the chain rule for the square root and trigonometric functions.

The derivative of -3√x is: - The derivative of √x is (1/2)x^(-1/2) [[1 #1]]. - Multiplying by -3 gives the derivative of -3√x as -3/2x^(-1/2).

The derivative of 1/3cos(x) is: - The derivative of cos(x) is -sin(x) [[2 #1]]. - Multiplying by 1/3 gives the derivative of 1/3cos(x) as -1/3sin(x).

The derivative of 1/2ctg(x) is: - The derivative of ctg(x) is -csc^2(x) [[3 #1]]. - Multiplying by 1/2 gives the derivative of 1/2ctg(x) as -1/2csc^2(x).

So, the derivative of the function y = -3√x + 1/3cos(x) - 1/2ctg(x) is: -3/2x^(-1/2) - 1/3sin(x) - 1/2csc^2(x).

2) Function: √x(-2x + 1)

To find the derivative of this function, we can use the product rule and the chain rule for the square root function.

The derivative of √x is: - The derivative of √x is (1/2)x^(-1/2) [[1 #1]].

Using the product rule, the derivative of √x(-2x + 1) is: (-2x + 1)*(1/2x^(-1/2)) + √x*(-2).

Simplifying this expression gives the derivative of the function as: -x^(-1/2) - √x.

3) Function: y = x/(x^2 - 1)

To find the derivative of this function, we can use the quotient rule.

The derivative of x/(x^2 - 1) is: ((x^2 - 1) - x*(2x))/(x^2 - 1)^2.

Simplifying this expression gives the derivative of the function as: (-x^2 - 1)/(x^2 - 1)^2.

These are the derivatives of the given functions. If you have any more questions or need further clarification, feel free to ask!