Computing Derivatives

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Derivatives of Elementary Functions

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Topics Derivatives of Elementary Functions

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[nx^n-1 + x^n-2Δx + ^... + Δx^n-1]
	=	anx^n-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) = a_nxⁿ + ^... + a₁x + a₀ is given by f (x) = na_nx^n-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) = Cr^t. Plugging into the formula for the derivative, we have

f'(t)	=
	=
	=
	=	Cr^t

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) = Cr^tlog(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) = Ce^t are the only functions that are equal to their own derivatives.