Geometrical Meaning of Derivative

The derivative also has a purely geometric meaning, quite apart from any algebraic formula. Take a curve $y=f(x)$ and a fixed point $P$ on it at $x=a$, so $P=(a,f(a))$. Pick a second point $Q$ on the same curve a little to the right, at $x=a+h$ for small $h>0$, so $Q=(a+h,f(a+h))$. The straight line joining $P$ and $Q$ is called a secant line, and its slope is $\text{slope of }PQ=\dfrac{f(a+h)-f(a)}{(a+h)-a}=\dfrac{f(a+h)-f(a)}{h}.$ Now imagine $h$ shrinking toward $0$: the point $Q$ slides back along the curve toward $P$, and the secant line's direction keeps rotating. In the limit, $Q$ merges into $P$ and the secant becomes the tangent line to the curve at $P$. Since $\lim_{h\to0}(\text{slope of secant }PQ)=\lim_{h\to0}\dfrac{f(a+h)-f(a)}{h}=f'(a),$ the slope of the tangent at $P$ is exactly $f'(a)$. The same limiting slope is reached whether $Q$ approaches from the right ($a+h$) or from the left, using a point $R=(a-h,f(a-h))$ instead. This is the geometrical meaning of the derivative: at any point $P(x_1,y_1)$ on the curve $y=f(x)$, the value $f'(x_1)$ is the slope of the tangent line to the curve at that point. …