Calculus 1B — The Derivative

The tangent line

continues from lesson 1 — values defined earlier in the course stay live here

ELI5: zoom far enough into any smooth curve and it looks straight — that straight line is the tangent, and its slope is the derivative. The tangent at x = a is the best straight-line stand-in for the curve nearby: y = f(a) + f′(a)·(x − a).

a_{tan} = 3

touch the parabola here

m_{tan} = 2 \cdot a_{tan}

6

f′(3) = 6 by the power rule

f_sq(a_tan) + m_tan * (x - a_tan)

the tangent line that kisses y = x² at x = 3 — change a_tan and it slides along

Real-world hook: engineers linearize near an operating point all the time — a sensor's slightly-curved response is treated as its tangent line for small signals, turning hard nonlinear math into easy straight-line math.

Try it yourself: what is the slope of y = x³ at x = 2? (It's f′(2) for the cube function.)

slope_{at2} = 12

✏️ Your turn: replace 0 with the slope of x³ at x = 2. We compare against the true slope there.

✓ pass

\mathrm{abs}\left(slope_{at2} - \frac{\mathrm{f_{cu}}\left(2 + 0.00001\right) - \mathrm{f_{cu}}\left(2 - 0.00001\right)}{2 \cdot 0.00001}\right) < 0.01

green when it matches the real steepness of x³ at x = 2