Answer to Question #231601 in Calculus for Mamaj

Question

1. Using the chain rule differentiate the following function.

a. y=(2x2+4x-2)2

b. y=(x2+2x-4)2

2. Using the product rule differentiate the following functions.

a. y=2x2-x2(1/2x2+5)2

b. y=(x+2)2(x+3)2

3. Using the quotient rule differentiate the following functions.

a. y=x2/(3x+5)2

b. y=e2x/x2+2

4. Differentiate the following functions.

a. y=x2e2x

b. y=ln(x+2x+1)

5. Differentiate the following functions (with respect of to x).

a. fx=2x2+3x(4x2-2)

b. fx=ln2x2+3e2+x

6. Differentiate the following functions and find the value of x.

a. y=x2+3x(x-2)

b. y=x2-3/x+2

Thank You

Answer 1

1.

a.

$$y'=((2x^2+4x^{-2})^2)'$$

$$=2(2x^2+4x^{-2})(2x^2+4x^{-2})'$$

$$=4(x^2+2x^{-2})(4x-8x^{-3})$$

$$=\dfrac{16(x^4+2)(x^4-2)}{x^5}=\dfrac{16(x^4-4)}{x^5}$$

b.

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

$$=2(x^2+2x-4)(x^2+2x-4)'$$

$$=2(x^2+2x-4)(2x+2)$$

$$=4(x+1)(x^2+2x-4)$$

2.

a.

$$y'=(2x^2-x^2(\dfrac{1}{2}x^2+5)^2)'$$

$$=4x-2x(\dfrac{1}{2}x^2+5)^2-2x^2 (\dfrac{1}{2}x^2+5)(\dfrac{1}{2}(2x)+5)$$

$$=4x-2x(\dfrac{1}{2}x^2+5)^2-x^3 (x^2+10)$$

$$=4x-\dfrac{1}{2}x^5-10x^3-50x-x^5-10x^3$$

$$=-\dfrac{3x^5+40x^3+92x}{2}$$

b.

$$y'=((x+2)^2(x+3)^2)'$$

$$=2(x+2)(x+3)^2+2(x+2)^2(x+3)$$

$$=2(x+2)(x+3)(x+3+x+2)$$

$$=2(x+2)(x+3)(2x+5)$$

3.

a.

$$y'=(\dfrac{x^2}{(3x+5)^2})'=\dfrac{2x(3x+5)^2-2x^2(3)(3x+5)}{(3x+5)^4}$$

$$=\dfrac{2x(3x+5-3x)}{(3x+5)^3}=\dfrac{10x}{(3x+5)^3}$$

b.

$$y'=(\dfrac{e^{2x}}{x^2+2})'=\dfrac{2e^{2x}(x^2+2)-2xe^{2x}}{(x^2+2)^2}$$

$$=\dfrac{2e^{2x}(x^2-x+2)}{(x^2+2)^2}$$

4.

a.

$$y'=(x^2e^{2x})'=2xe^{2x}+2x^2e^{2x}=2x(x+1)e^{2x}$$

b.

$$y'=(\ln(x+2x+1))'=\dfrac{3}{3x+1}$$

$$y'=(\ln((x+2)(x+1))'=\dfrac{x+1+x+2}{(x+2)(x+1)}$$

$$=\dfrac{2x+3}{(x+2)(x+1)}$$

5.

a.

$$f'(x)=(2x^2+3x(4x^2-2))'$$

$$=(12x^3+2x^2-6x)'=36x^2+4x-6$$

b.

$$f'(x)=(\ln^2(x^2)+3e^{2+x})'$$

$$=2\ln(x^2)(\dfrac{1}{x^2})(2x)+3e^{2+x}$$

$$=\dfrac{8\ln|x|}{x}+3e^{2+x}$$

6.

a.

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

$$=8x-6, x\in \R$$

b.

$$y'=(x^2-\dfrac{3}{x+2})'=2x-\dfrac{3}{(x+2)^2}, x\not=-2$$