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. …

Figure 7.1A straight line l in the XY-plane with a horizontal 'Change in x' leg and a vertical 'Change in y' leg forming a right triangle, illustrating that slope is the constant ratio of vertical change to horizontal change.

Fig. 7.1 — A straight line l in the XY-plane with a horizontal 'Change in x' leg and a vertical 'Change in y' leg forming a right triangle, illustrating that slope is the constant ratio of vertical change to horizontal change.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. A straight line l in the XY-plane with a horizontal 'Change in x' leg and a vertical 'Change in y' leg forming a right triangle, illustrating that slope is the constant ratio of vertical chang …

Figure 7.2The curve y = f(x) with its tangent line at the point (x, y), the angle theta the tangent makes with the x-axis, showing the slope of a curve equals tan(theta).

Fig. 7.2 — The curve y = f(x) with its tangent line at the point (x, y), the angle theta the tangent makes with the x-axis, showing the slope of a curve equals tan(theta).

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. The curve y = f(x) with its tangent line at the point (x, y), the angle theta the tangent makes with the x-axis, showing the slope of a curve equ …

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