Answer to Question #348965 in Calculus for Oodi

Question

Part I. Differentiation Rules on Algebraic Functions

Apply the appropriate differentiation technique/s to determine the derivative of the following algebraic functions.

1. f(x)=x^(-6)+4x^3-10
2. f(x)=5x^3-4/x^2
3. y=√(5-2x)
4. g(x)=(x^2-5x-7)(x^2-5)
5. g(x)=(3x^2-4x+1)(2x-3)
6. h(x)=(4x^3-2x+1)/(2x-3)
7. h(x)=(4x+3)/(5x^2-3x+6 )
8. y=(3x^2+2x-3)^3
9. y=(5x+3)^2 (2x-1)
10. y=(3x-2)^2/(4x+1)

Part II. Differentiation Rules on Trigonometric Functions and Higher-Order Derivatives

Apply the appropriate differentiation technique/s to determine the derivative of the following trigonometric functions.

1. y=cos〗⁡(x^2+3)
2. y=sin⁡〖(5x^2-7)〗
3. y=cot⁡(6x)
4. y=tan⁡(cos⁡x )
5. y=csc⁡(5x^3 )

Find the first derivative, second derivative, and third derivative for each of the following functions.

1. y=6x^4+3x^3-4x^2-1
2. f(x)=(2x+3)^4

Answer 1

Part I.

1.

$$f'(x)=-6x^{-7}+12x^2$$

2.

$$f'(x)=15x^2+8/x^3$$

3.

$$y'=-\dfrac{1}{\sqrt{5-2x}}$$

4.

$$g'(x)=(2x-5)(x^2-5)+(x^2-5x-7)(2x)$$

$$=2x^3-10x-5x^2+25+2x^3-10x^2-14x$$

$$=4x^3-15x^2-24x+25$$

5.

$$g'(x)=(6x-4)(2x-3)+2(3x^2-4x+1)$$

$$=12x^2-18x-8x+12+6x^2-8x+2$$

$$=18x^2-34x+14$$

6.

$$h'(x)=\dfrac{(12x^2-2)(2x-3)-2(4x^3-2x+1)}{(2x-3)^2}$$

$$=\dfrac{24x^3-36x^2-4x+6-8x^3+4x-2}{(2x-3)^2}$$

$$=\dfrac{16x^3-36x^2+4}{(2x-3)^2}$$

7.

$$h'(x)=\dfrac{4(5x^2-3x+6)-(10x-3)(4x+3)}{(5x^2-3x+6)^2}$$

$$=\dfrac{20x^2-12x+24-40x^2-30x+12x+9}{(5x^2-3x+6)^2}$$

$$=\dfrac{-20x^2-30x+33}{(5x^2-3x+6)^2}$$

8.

$$y'=3(3x^2+2x-3)^2(6x+2)$$

$$=6(3x+1)(3x^2+2x-3)^2$$

9.

$$y'=2(5x+3)(5)(2x-1)+2(5x+3)^2$$

$$=2(5x+3)(10x-5+5x+3)$$

$$=2(5x+3)(15x-2)$$

10.

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

$$=\dfrac{2(3x-2)(12x+3-6x+4)}{(4x+1)^2}$$

$$=\dfrac{2(3x-2)(6x+7)}{(4x+1)^2}$$

Part II.

1.

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

2.

$$y'=10x\cos(5x^2-7)$$

3.

$$y'=-\dfrac{6}{\sin^2(6x)}=-6\csc^2(6x)$$

4.

$$y'=-\dfrac{\sin x}{\cos^2(\cos x)}=-\sin x\sec^2 (\cos x)$$

5.

$$y'=-\dfrac{1}{\sin^2(5x^3)}(\cos (5x^3))(15x^2)$$

$$=-15x^2\cot(5x^3)\csc x(5x^3)$$

Part III.

1.

$$y'=24x^3+9x^2-8x$$

$$y''=72x^2+18x-8$$

$$y'''=144x+18$$

2.

$$f'(x)=8(2x+3)^3$$

$$f''(x)=48(2x+3)^2$$

$$f'''(x)=192(2x+3)$$