Differential Calculus Explorer

A limit $\lim_{x\to c} f(x) = L$ means: for every $\varepsilon > 0$ you name, I can supply a $\delta > 0$ so that whenever $0 < |x-c| < \delta$, we have $|f(x) - L| < \varepsilon$. Pick a function, set a tolerance $\varepsilon$ with the slider, and the visualisation finds the largest strip width $\delta$ that works.

The derivative of $f$ at $a$ is the limit of secant slopes as the second point slides in. Choose a function and a base point $a$; shrink $h$ towards zero and watch the secant line rotate onto the tangent line, while the numerical slope converges to $f'(a)$.

The four workhorse rules let us differentiate almost anything built from elementary pieces. Pick an example and overlay the plot of $f$ with $f'$ in a contrasting colour to see how the rule plays out visually.

When a quotient $f/g$ hits an indeterminate form $0/0$ or $\infty/\infty$ at $x=c$, the limit often equals the limit of $f'/g'$ instead. Pick an example and compare the two ratios as $x$ approaches the singular point.

Critical points sit at $f'(x) = 0$, inflection points at $f''(x) = 0$. The sign of $f'$ tells you whether $f$ is rising or falling; the sign of $f''$ tells you whether it is concave up or down. Regions where $f'>0$ are shaded green; regions where $f'<0$ are shaded pink.

Near a point $a$, the best linear approximation of $f$ is $L(x) = f(a) + f'(a)(x-a)$. The error $f(x) - L(x)$ vanishes faster than $x-a$ as $x \to a$. Drag the base point and watch how the linearisation tracks the function locally — and how quickly the approximation error grows away from $a$.