Homework: Quotient Rule – Informal Calculus

28 Homework: Quotient Rule

Find the derivatives of the following functions.
1. $f (x) = \frac{e^{x}}{x}$
  
  $f^{'} (x) = \frac{x e^{x} - e^{x}}{x^{2}}$
  
  ans
2. $g (x) = \frac{\sqrt[3]{x}}{\ln (x)}$
  
  $g^{'} (x) = \frac{\frac{1}{3} x^{- 2 / 3} \ln (x) - \sqrt[3]{x} / x}{(\ln x)^{2}} = \frac{3 x^{- 2 / 3} (\ln (x) - 1)}{3 (\ln x)^{2}}$
  
  ans
3. $h (x) = \frac{1}{x^{2} + 5 x + 6}$
  
  $h^{'} (x) = \frac{- (2 x + 5)}{(x^{2} + 5 x + 6)^{2}}$
  
  ans
4. $i (x) = \frac{\cos (x)}{1 + x^{2}}$
  
  $i^{'} (x) = \frac{- (1 + x^{2}) \sin (x) - 2 x \cos (x)}{(1 + x^{2})^{2}}$
  
  ans
5. $j (x) = \frac{\ln (x)}{x^{2}}$
  
  $j^{'} (x) = \frac{1 - 2 \ln (x)}{x^{3}}$
  
  ans
6. $k (x) = e^{- x}$ (How can you write this as a fraction?)
  
  $k^{'} (x) = \frac{- e^{x}}{(e^{x})^{2}} = - \frac{1}{e^{x}}$
  
  ans
7. $ℓ (x) = \frac{x e^{x}}{1 + x}$
  
  $ℓ^{'} (x) = \frac{x^{2} e^{x} + x e^{x} + e^{x}}{(1 + x)^{2}}$
  
  ans

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