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The Chain Rule is a formula for differentiating composite functions — functions of the form h(x) = f(g(x)). It is one of the most important rules in calculus and is essential for differentiating all but the simplest functions. The rule states: h'(x) = f'(g(x)) × g'(x).

الصيغة

In Leibniz notation: dy/dx = (dy/du) · (du/dx)
In
(dy/du) · (du/dx) — (dy/du) · (du/dx)

دليل خطوة بخطوة

  1. 1If h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x)
  2. 2In Leibniz notation: dy/dx = (dy/du) · (du/dx)
  3. 3Identify the outer function f and inner function g
  4. 4Differentiate the outer function with respect to the inner, then multiply by the derivative of the inner

أمثلة محلولة

الإدخال
h(x) = sin(x²), outer=sin, inner=x²
النتيجة
h'(x) = cos(x²) · 2x
Outer derivative: cos(u); inner derivative: 2x
الإدخال
h(x) = (3x+1)⁵
النتيجة
h'(x) = 5(3x+1)⁴ · 3 = 15(3x+1)⁴
الإدخال
h(x) = e^(x²)
النتيجة
h'(x) = e^(x²) · 2x
الإدخال
h(x) = ln(cos(x))
النتيجة
h'(x) = (1/cos(x)) · (−sin(x)) = −tan(x)

أسئلة شائعة

What is Chain Rule?

The Chain Rule is a formula for differentiating composite functions — functions of the form h(x) = f(g(x)). It is one of the most important rules in calculus and is essential for differentiating all but the simplest functions

How accurate is the Chain Rule calculator?

The calculator uses the standard published formula for chain rule. Results are accurate to the precision of the inputs you provide. For financial, medical, or legal decisions, always verify with a qualified professional.

What units does the Chain Rule calculator use?

This calculator works with inches. You can enter values in the units shown — the calculator handles all conversions internally.

What formula does the Chain Rule calculator use?

The core formula is: If h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x). Each step in the calculation is shown so you can verify the result manually.

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