# 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))

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## 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 \( f'(x)=\frac{sin(x)}{cos(x)} \)  
   \[ f'(x)=tan(x) \]