Applications of derivatives

When a function $y=f(x)$ is differentiable, its derivative $f^{\prime }(x)$ tells how $y$ changes with $x$. The slope of the tangent line at a point $x=a$ is exactly $f^{\prime }(a)$. To write the tangent line, use the point–slope form $y-f(a)=f^{\prime }(a)(x-a)$. This same derivative idea lets us locate where a function stops increasing or decreasing (critical points where $f^{\prime }(x)=0$) and where its curvature changes (inflection points where $f^{\prime \prime }(x)=0$). In optimization problems we set $f^{\prime }(x)=0$ to find candidates for maxima or minima, then test them. In related‑rates problems we differentiate an equation linking several quantities with respect to time, replacing $dx/dt$, $dy/dt$, etc., by the known rates and solving for the unknown one.