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Problem : Compute the derivative of f (x) = 6x4 +8x3 +12x2 + 24x.

We differentiate term-by-term to get


f'(x)=(4)6x3 + (3)8x2 + (2)12x + 24  
 =24(x3 + x2 + 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) = - x2 + 3. For which values of x does h(x) = h'(x)?

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

- x2 + 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) = - x2 + 3 and h'(x) = - 2x
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