Derivative (examples)

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For more background on this topic, see derivative.

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[edit] Example 1

Consider f(x) = 5:

f'(x)=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} = \lim_{h\rightarrow 0} \frac{f(x+h)-5}{h} =  \lim_{h\rightarrow 0} \frac{(5-5)}{h} = \lim_{h\rightarrow 0} \frac{0}{h} = \lim_{h\rightarrow 0} 0 = 0

The derivative of a constant function is zero.

[edit] Example 2

Consider the graph of f(x) = 2x − 3. If the reader has an understanding of algebra and the Cartesian coordinate system, the reader should be able to independently determine that this line has a slope of 2 at every point. Using the above quotient (along with an understanding of the limit, secant, and tangent) one can determine the slope at (4,5):

\begin{align} f'(4) &= \lim_{h\to 0}\frac{f(4+h)-f(4)}{h}           \\       &= \lim_{h\to 0}\frac{2(4+h)-3-(2\cdot 4-3)}{h} \\       &= \lim_{h\to 0}\frac{8+2h-3-8+3}{h}            \\       &= \lim_{h\to 0}\frac{2h}{h}                    \\       &= 2 \end{align}

The derivative and slope are equivalent.

[edit] Example 3

Via differentiation, one can find the slope of a curve. Consider f(x) = x2:

f'(x)\, = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}
= \lim_{h\rightarrow 0}\frac{(x+h)^2 - x^2}{h}
= \lim_{h\rightarrow 0}\frac{x^2 + 2xh + h^2 - x^2}{h}
= \lim_{h\rightarrow 0}\frac{2xh + h^2}{h}
= \lim_{h\rightarrow 0}(2x + h) = 2x

For any point x, the slope of the function f(x) = x2 is f'(x) = 2x.

[edit] Example 4

Consider f(x) = \sqrt{x}:

f'(x)\, = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}
= \lim_{h\rightarrow 0}\frac{\sqrt{x+h} - \sqrt{x}}{h}
= \lim_{h\rightarrow 0}\left(\frac{\sqrt{x+h} - \sqrt{x}}{h}\right) \left(\frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}}\right)
= \lim_{h\rightarrow 0}\frac{x+h - x}{h(\sqrt{x+h} + \sqrt{x})}
= \lim_{h\rightarrow 0}\frac{1}{\sqrt{x+h} + \sqrt{x}}
= \frac{1}{2 \sqrt{x}}

[edit] Example 5

The same as the previous example, but now we search the derivative of the derivative.
Consider f(x) = \sqrt{x}:

\begin{align} f''(x) &= \lim_{h\to 0}\frac{f'(x+h)-f'(x)}{h} \\        &= \lim_{h\to 0} \frac{\frac{1}{2 \sqrt{x+h}}-\frac{1}{2 \sqrt{x}}}{h} \\        &= \lim_{h\to 0} \frac{\left(\frac{1}{2 \sqrt{x+h}}-\frac{1}{2 \sqrt{x}}\right)(2 \sqrt{x+h}+2 \sqrt{x})}{h(2 \sqrt{x+h}+2 \sqrt{x})} \\        &= \lim_{h\rightarrow 0} \frac{\frac{2 \sqrt{x}}{2 \sqrt{x+h}}-\frac{2 \sqrt{x+h}}{2 \sqrt{x}}}{h(2 \sqrt{x+h}+2 \sqrt{x})} \\        &= -\frac{1}{4 x \sqrt{x}} \end{align}
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