Master the Chain Rule for differentiation in CBSE Class 11 Applied Mathematics. Learn to differentiate composite functions with step-by-step examples.
A function of a function, or composite function, is defined as f(g(x)). To differentiate such functions, we cannot simply differentiate the outer function; we must account for the rate of change of the inner function. This process is known as the Chain Rule.
Step 1: Let u = 3x + 5, so y = u⁴.
Step 2: Find dy/du = 4u³ and du/dx = 3.
Step 3: Multiply: dy/dx = (dy/du) · (du/dx) = 4(3x + 5)³ · 3.
Answer: 12(3x + 5)³
When a function is raised to a power, the Chain Rule simplifies to the Generalized Power Rule. We differentiate the outer power first, keeping the inner function constant, and then multiply by the derivative of the inner function.
Step 1: Identify u = x² + 1, n = 3.
Step 2: Apply formula: 3(x² + 1)² · d/dx(x² + 1).
Step 3: Calculate derivative of inner: 3(x² + 1)² · 2x.
Answer: 6x(x² + 1)²
The Chain Rule is essential for differentiating complex algebraic expressions involving roots or denominators. By rewriting radicals as fractional exponents, we can apply the power rule in conjunction with the Chain Rule.
Step 1: Rewrite as y = (2x² + 3)¹/².
Step 2: Apply power rule: 1/2(2x² + 3)⁻¹/² · d/dx(2x² + 3).
Step 3: Multiply by inner derivative 4x: (1/2) · (1/√(2x² + 3)) · 4x.
Answer: 2x / √(2x² + 3)