Derivative at a point & tangent lines

Japanese school year: Math II

What you learn

This topic covers the geometric interpretation of the derivative as the slope of a curve, enabling students to construct tangent line equations. It is widely used in linear approximation and analyzing the local directional behavior of curves. Familiarity with polynomial differentiation and line equations from coordinate geometry is required.

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Key points

This formula defines the derivative $f'(a)$ , which gives the instantaneous rate of change at $x = a$ . Geometrically, it represents the slope of the tangent line at $(a, f(a))$ .

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

This formula gives the equation of the tangent line touching the curve at $(a, f(a))$ . Here, the derivative $f'(a)$ acts as the slope of the line.

y - f(a) = f'(a)(x - a)

This formula finds the normal line, which is perpendicular to the tangent at $(a, f(a))$ . Because perpendicular slopes multiply to $-1$ , the slope of the normal line is $-\frac{1}{f'(a)}$ .

y - f(a) = -\frac{1}{f'(a)}(x - a) \quad (f'(a) \neq 0)

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Practice 1Coming soon