Chain rule: a box inside a boxश्रृंखला नियम: डिब्बे के अंदर डिब्बाChain rule: dabbe ke andar dabba

NCERT Theorem 4 · Class 12 Maths · Chapter 5 · interactive diagram

Parts and what they mean

• Input x (इनपुट x) — x goes into the inner box first.
• Inner function t = u(x) = x^2 (भीतरी फलन t = u(x) = x^2) — Works first; its output t goes into the outer box.
• Outer function v(t) = \sin t (बाहरी फलन v(t) = \sin t) — Works last: f = v \circ u, so f(x) = \sin(x^2).
• \frac{dt}{dx} = 2x (\frac{dt}{dx} = 2x) — How fast the inner box output changes with x.
• \frac{dv}{dt} = \cos t (\frac{dv}{dt} = \cos t) — How fast the outer box output changes with t.
• Output f(x) = \sin(x^2) (आउटपुट f(x) = \sin(x^2)) — The composite function.
• Chain rule result (श्रृंखला नियम का परिणाम) — \frac{df}{dx} = \frac{dv}{dt} \cdot \frac{dt}{dx} = \cos(x^2) \cdot 2x (Example 21).

Original diagram drawn by Simple Learning Hub, based on NCERT Theorem 4. Report a mistake