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In studying polynomial functions, it is therefore enough to find the derivative of a monomial function of the form f (x) = axn. Plugging into the formula for the derivative, we have


f'(x)=  
 =  
 =  
 =a[nxn-1 + xn-2Δx + ... + Δxn-1]  
 =anxn-1  

Thus, to take the derivative of a monomial function, we multiply by the exponent and reduce the exponent by 1. Using the property of the derivative mentioned above, we see that the derivative of the polynomial function f (x) = anxn + ... + a1x + a0 is given by f (x) = nanxn-1 + ... + a2x + a1.

We will wait until we have the quotient rule at our disposal before we calculate the derivatives of rational functions.

Derivatives of Power Functions

A power function has the form f (t) = Crt. Plugging into the formula for the derivative, we have


f'(t)=  
 =  
 =  
 =Crt  

The limit in the final expression above does not depend on t, so it is a constant. In fact, this limit is one way of defining the value of the natural logarithm function at r, or log(r). Thus we have

f'(t) = Crtlog(r)    

In the special case where r = e, where e is the number such that log(e) = 1, we have f'(t)=f(t). The functions f (t) = Cet are the only functions that are equal to their own derivatives.

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