Question

Найдите производную функции : 1) 3/ x+2 корень из х - е в степени x2) (3х-5) в степени 43) 3 sin2xcosx4) х в степени 3/ х в степени 2+5 Помогите пожалуйста решить

Answer 1

1) 
$\displaystyle y`=( \frac{3}{x}+2 \sqrt{x} -e^x)`=3(x^{-1})`+2(x^{1/2})`-(e^x)`=\\\\=- \frac{3}{x^2}+ \frac{1}{ \sqrt{x} }-e^x$

2) 
$\displaystyle y`=((3x-5)^4)`=4(3x-5)^3*3=12(3x-5)^3$

3) 
$\displaystyle y`=(3sin2x*cosx)`=3((sin2x)`*cosx+(cosx)`*sin2x)=\\\\=3(2cos2x*cosx-sinx*sin2x)$

4) 
$\displaystyle y`=( \frac{x^3}{x^2+5})`= \frac{(x^3)`*(x^2+5)-(x^2+5)`*x^3}{(x^2+5)^2}=\\\\= \frac{3x^2(x^2+5)-2x*x^3}{(x^2+5)^2}= \frac{x^4+15x^2}{(x^2+5)^2}$

Answer 2

Derivative of Functions

1) The derivative of the function \( f(x) = \frac{3}{x} + 2\sqrt{x} - e^{x^2} \) is: - The derivative of \( \frac{3}{x} \) is \( -\frac{3}{x^2} \) [[1]]. - The derivative of \( 2\sqrt{x} \) is \( \frac{1}{\sqrt{x}} \) which can be written as \( 2x^{-\frac{1}{2}} \) [[2]]. - The derivative of \( e^{x^2} \) is \( 2xe^{x^2} \) using the chain rule [[3]].

Therefore, the derivative of the function \( f(x) = \frac{3}{x} + 2\sqrt{x} - e^{x^2} \) is: \[ f'(x) = -\frac{3}{x^2} + 2x^{-\frac{1}{2}} - 2xe^{x^2} \]

2) The derivative of the function \( g(x) = (3x-5)^{4} \) is: - Using the chain rule, the derivative of \( (3x-5)^{4} \) is \( 4(3x-5)^{3}(3) \) [[4]].

Therefore, the derivative of the function \( g(x) = (3x-5)^{4} \) is: \[ g'(x) = 12(3x-5)^{3} \]

3) The derivative of the function \( h(x) = 3\sin^2(x)\cos(x) \) is: - Using the product rule and chain rule, the derivative of \( 3\sin^2(x)\cos(x) \) is \( 3\cos(x)\sin^2(x) - 3\sin^3(x) \) [[5]].

Therefore, the derivative of the function \( h(x) = 3\sin^2(x)\cos(x) \) is: \[ h'(x) = 3\cos(x)\sin^2(x) - 3\sin^3(x) \]

4) The derivative of the function \( k(x) = x^3/x^2 \) is: - Simplifying \( x^3/x^2 \) gives \( x \). - Therefore, the derivative of the function \( k(x) = x^3/x^2 \) is: \[ k'(x) = 1 \]

5) The derivative of the function \( m(x) = x^{3}/x^{2} + 5 \) is: - Using the power rule, the derivative of \( x^{3}/x^{2} \) is \( x^{3-2} = x \). - Therefore, the derivative of the function \( m(x) = x^{3}/x^{2} + 5 \) is: \[ m'(x) = x + 0 = x \]

I hope this helps! If you have any more questions or need further assistance, feel free to ask.