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Derivatives From the Limit Definition

April 9th, 2009 |

Author: Merrin

The differentiation rules we develop in this chapter will obscure the use of the limit, but we still need the definition to derive the differentiation rules. If you are a glutton for punishment or are encountering a new function for the first time, deriving the derivative from the limit may be the way to go.

Example 1. Find the derivative of square root of x from the limit definition of the derivative.

Solution 1.We use a conjugate expression to find this derivative

$f'(x)=\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{\sqrt{x+\Delta x} -\sqrt{x} }{\Delta x}$

$f'(x)=\mathop{\lim }\limits_{\Delta x\to 0}\left( \cfrac{\sqrt{x+\Delta x} -\sqrt{x} }{\Delta x} \right)\left( \cfrac{\sqrt{x+\Delta x} +\sqrt{x} }{\sqrt{x+\Delta x} +\sqrt{x}} \right)$

$f'(x)=\mathop{\lim }\limits_{\Delta x\to 0}\cfrac{\Delta x }{\Delta x (\sqrt{x}+\sqrt{x+\Delta x})}=\cfrac{1}{2\sqrt{x}}$

Example 2. Find the derivative of

$f(x)=x^{2}$

Solution 2. Apply the definition of the derivative

${\cfrac{d}{dx} x^{2} } = {\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{(x+\Delta x)^{2} -x^{2} }{\Delta x} =\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{2x\Delta x+(\Delta x)^{2} }{\Delta x} } \\ = {2x+\mathop{\lim }\limits_{\Delta x\to 0} \Delta x=2x}$

The slope of a parabola is zero at its vertex as we can see from f’(0)=0 and for other points along the curve the slope is f’(x)=2x.

Both these problems are special case of the power rule with n=1/2 and

n=2. Here is a slightly more complicated example.

Example 3. Find the derivative of

$x^{1/4}$

using the definition of the derivative.

Solution 3. Recall our discussion of conjugate expressions in relation to limits from previous sections. We will use the fact that

$a^4-b^4=(a-b)(a^3+a^2b+ab^2+b^3)$

Plugging in in the definition of the limit for this function we have

$\cfrac{d}{dx} x^{1/4} = \mathop{\lim }\limits_{\Delta x\to 0} \cfrac{(x+\Delta x)^{1/4} -x^{1/4} }{\Delta x} \\= \mathop{\lim }\limits_{\Delta x\to 0} \cfrac{\Delta x}{\Delta x} \cfrac{1}{(x+\Delta x)^{3/4} + (x+\Delta x)^{2/4}x^{1/4}+(x+\Delta x)^{1/4}x^{2/4}+x^{3/4}}\\= \cfrac{1}{4x^{3/4}}$