Question #348965

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
1
Expert's answer
2022-06-08T17:08:01-0400

Part I.

1.

f(x)=6x7+12x2f'(x)=-6x^{-7}+12x^2

2.

f(x)=15x2+8/x3f'(x)=15x^2+8/x^3

3.

y=152xy'=-\dfrac{1}{\sqrt{5-2x}}

4.

g(x)=(2x5)(x25)+(x25x7)(2x)g'(x)=(2x-5)(x^2-5)+(x^2-5x-7)(2x)


=2x310x5x2+25+2x310x214x=2x^3-10x-5x^2+25+2x^3-10x^2-14x

=4x315x224x+25=4x^3-15x^2-24x+25

5.

g(x)=(6x4)(2x3)+2(3x24x+1)g'(x)=(6x-4)(2x-3)+2(3x^2-4x+1)


=12x218x8x+12+6x28x+2=12x^2-18x-8x+12+6x^2-8x+2

=18x234x+14=18x^2-34x+14

6.

h(x)=(12x22)(2x3)2(4x32x+1)(2x3)2h'(x)=\dfrac{(12x^2-2)(2x-3)-2(4x^3-2x+1)}{(2x-3)^2}


=24x336x24x+68x3+4x2(2x3)2=\dfrac{24x^3-36x^2-4x+6-8x^3+4x-2}{(2x-3)^2}

=16x336x2+4(2x3)2=\dfrac{16x^3-36x^2+4}{(2x-3)^2}

7.

h(x)=4(5x23x+6)(10x3)(4x+3)(5x23x+6)2h'(x)=\dfrac{4(5x^2-3x+6)-(10x-3)(4x+3)}{(5x^2-3x+6)^2}


=20x212x+2440x230x+12x+9(5x23x+6)2=\dfrac{20x^2-12x+24-40x^2-30x+12x+9}{(5x^2-3x+6)^2}

=20x230x+33(5x23x+6)2=\dfrac{-20x^2-30x+33}{(5x^2-3x+6)^2}

8.


y=3(3x2+2x3)2(6x+2)y'=3(3x^2+2x-3)^2(6x+2)

=6(3x+1)(3x2+2x3)2=6(3x+1)(3x^2+2x-3)^2

9.


y=2(5x+3)(5)(2x1)+2(5x+3)2y'=2(5x+3)(5)(2x-1)+2(5x+3)^2

=2(5x+3)(10x5+5x+3)=2(5x+3)(10x-5+5x+3)

=2(5x+3)(15x2)=2(5x+3)(15x-2)

10.

y=6(3x2)(4x+1)4(3x2)2(4x+1)2y'=\dfrac{6(3x-2)(4x+1)-4(3x-2)^2}{(4x+1)^2}


=2(3x2)(12x+36x+4)(4x+1)2=\dfrac{2(3x-2)(12x+3-6x+4)}{(4x+1)^2}

=2(3x2)(6x+7)(4x+1)2=\dfrac{2(3x-2)(6x+7)}{(4x+1)^2}

Part II.

1.

y=2xsin(x2+3)y'=-2x\sin(x^2+3)

2.


y=10xcos(5x27)y'=10x\cos(5x^2-7)

3.


y=6sin2(6x)=6csc2(6x)y'=-\dfrac{6}{\sin^2(6x)}=-6\csc^2(6x)

4.


y=sinxcos2(cosx)=sinxsec2(cosx)y'=-\dfrac{\sin x}{\cos^2(\cos x)}=-\sin x\sec^2 (\cos x)

5.


y=1sin2(5x3)(cos(5x3))(15x2)y'=-\dfrac{1}{\sin^2(5x^3)}(\cos (5x^3))(15x^2)

=15x2cot(5x3)cscx(5x3)=-15x^2\cot(5x^3)\csc x(5x^3)

Part III.

1.


y=24x3+9x28xy'=24x^3+9x^2-8x

y=72x2+18x8y''=72x^2+18x-8

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

2.


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

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

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


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