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Computing Derivatives

Problems

Derivatives of Elementary Functions

Techniques of Differentiation

Problem : Compute the derivative of f (x) = 6x 4 +8x 3 +12x 2 + 24x .

We differentiate term-by-term to get


f'(x) = (4)6x 3 + (3)8x 2 + (2)12x + 24  
  = 24(x 3 + x 2 + x + 1)  

Problem : What is the second derivative of g(x) = sin(x) ?

We have


g'(x) = cos(x)  
g''(x) = -sin(x)  

Problem : Let h(x) = - x 2 + 3 . For which values of x does h(x) = h'(x) ?

Since h'(x) = - 2x , we want to solve for x in the equation

- x 2 + 3 = - 2x    

or equivalently,

0 = (x + 1)(x - 3)    

Thus h(x) = h'(x) for x = - 1 and for x = 3 .
Figure %: Plot of h(x) = - x 2 + 3 and h'(x) = - 2x

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