Calculus 1 Worksheet: Derivatives

Instructions

Find the derivative of each function with respect to x. Show all your steps.

Problems

1. Power Rule
   Find f'(x) given:
   f(x)=4x^{5}-3x^{3}+7x-2

2. Product Rule
   Find f'(x) given:
   f(x)=x^{2}e^{x}

3. Quotient Rule
   Find f'(x) given:
   f(x)=\frac{3x+1}{x^{2}-2}

4. Chain Rule (Trigonometric/Logarithmic)
   Find f'(x) given:
   f(x) = ln(cos(x))

5. Chain Rule (Polynomial)
   Find f'(x) given:
   f(x)=ln(cos(x))

Solutions

1. Solution for: f(x) = 4x^{5} - 3x^{3} + 7x - 2
   Using the Power Rule:

   [ f'(x)=\frac{d}{dx}(4x^{5}) - \frac{d}{dx}(3x^{3}) + \frac{d}{dx}(7x) - \frac{d}{dx}(2) ]
   [ f'(x)=4(5x^{4}) - 3(3x^{2}) + 7(1) - 0 ]
   [ f'(x)=20x^{4} - 9x^{2} + 7 ]

2. Solution for: f(x) = x^{2}e^{x}
   Using the Product Rule:

   Let ( u=x^{2} ) and ( v=e^{x} ):

   • ( u' = 2x )
   • ( v' = e^{x} )
     [ f'(x)=(u'v + uv') = (2x)(e^{x}) + (x^{2})(e^{x}) ]
     [ f'(x)=e^{x}(2x + x^{2}) ]

3. Solution for: f(x) = \frac{3x+1}{x^{2}-2}
   Using the Quotient Rule:

   Let ( u=3x+1 ) and ( v=x^{2}-2 ):
   [ f'(x)=\frac{u'v - uv'}{v^{2}} ]

   • ( u' = 3 )
   • ( v' = 2x )
     [ f'(x)=\frac{(3)(x^{2}-2) - (3x+1)(2x)}{(x^{2}-2)^{2}} ]
     [ f'(x)=\frac{3x^{2}-6 - (6x^{2}+2x)}{(x^{2}-2)^{2}} ]

4. Solution for: f(x) = 2x^{3} + 5
   Using the Chain Rule:
   Let ( u=2x^{3}+5 ) then ( y=u^{4} )
   [ f'(x)=4(2x^{3}+5)^{3} \cdot (6x^{2}) ]
   [ f'(x)=24x^{2}(2x^{3}+5)^{3} ]

5. Solution for: f(x) = ln(cos(x))
   Using the Chain Rule:
   Let ( u=cos(x) ) then ( f'(x)=\frac{sin(x)}{cos(x)} )
   [ f'(x)=tan(x) ]