Mathematics · Ch 7 — Applications of Differential Calculus

Derivative as Slope

7.2.1

Derivative as Slope

Slope of a line. For a non-vertical line $l$ , take any horizontal segment starting on $l$ and the vertical segment from its end back to $l$ ; the ratio (vertical length)/(horizontal length) is always the same constant, called the slope $m$ of the line. A line is increasing if $m>0$ , decreasing if $m<0$ , and constant (horizontal) if $m=0$ . The equation $y=mx+c$ is exactly the line of slope $m$ and $y$ -intercept $c$ .

Slope of a curve. For a curve $y=f(x)$ , the line joining $(x,f(x))$ and $(x+h,f(x+h))$ has slope

\frac{f(x+h)-f(x)}{h}\qquad\text{(the Newton quotient).}

Letting $h\to0$ gives the slope of the curve at the point $(x,y)$ itself:

\lim_{h\to0}\frac{f(x+h)-f(x)}{h}=f'(x)\qquad\text{(the limit of the Newton quotient).}

Note

If $\theta$ is the angle the tangent to $y=f(x)$ at $(x,y)$ makes with the $x$ -axis (measured anticlockwise), then $f'(x)=\tan\theta$ . Because of this, $f'(x)$ is also written $\dfrac{dy}{dx}$ and called the instantaneous rate of change; the average rate of change over an interval is the ordinary Newton quotient (a chord slope) over that interval. …

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