Question

Найдите производную функций: 1.y=(x^2+2x+4)*(x-2)? 2. y= cosx*ctgx? 3.y=x^2 / 3-4x? 4. y=x / x^2+1?

Answer 1

1)$y'=(2x+2)(x-2)+(x^2+2x+4)= \\ \\ =2x^2-4x+2x-4+x^2+2x+4=3x^2$

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2) $y= cosx\frac{cosx}{sinx} = \frac{cos^2x}{sinx} = \frac{1-sin^2x}{sinx} = \frac{1}{sinx}-sinx \\ \\ y'= -\frac{cosx}{sin^2x}-cosx =-cosx(1+ \frac{1}{sin^2x})$

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3)$y= \frac{x^2}{3-4x} \\ \\ y'= \frac{2x(3-4x)+4x^2}{(3-4x)^2} = \frac{6x-8x^2-4x^2}{(3-4x)^2} = \frac{6x(1-2x)}{(3-4x)^2}$

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