Mastering Continuity and Differentiability Ex 5.2: The Chain Rule

Understanding the Chain Rule for Composite Functions

The Chain Rule is a fundamental theorem in differential calculus that allows us to find the derivative of a composite function. A composite function is essentially a 'function of a function', meaning one function is nested inside another. For instance, if you have y = sin(x^2), here sin(u) is the outer function and u = x^2 is the inner function. Without the Chain Rule, differentiating such expressions would be extremely difficult or even impossible using basic rules.

The Chain Rule states that if a function y = f(u) where u = g(x), then the derivative of y with respect to x is given by:

dy/dx = (dy/du) * (du/dx)

In simpler terms, you differentiate the 'outer' function first, treating the 'inner' function as a single variable. Then, you multiply this result by the derivative of the 'inner' function with respect to x. This process can be extended for functions composed of three or more functions. For example, if y = f(g(h(x))), then dy/dx = f'(g(h(x))) g'(h(x)) h'(x). This systematic approach makes complex differentiation problems manageable and is a cornerstone of calculus, frequently appearing in physics, engineering, and economics problems. Mastering this rule is key to scoring well in this chapter.

Key Concepts and Definitions

Composite Function

A function that is formed by combining two or more functions, where the output of one function becomes the input of another. It's often written as f(g(x)) or (f ∘ g)(x).

Differentiability

A function is differentiable at a point if its derivative exists at that point. Geometrically, this means the function has a well-defined, non-vertical tangent line at that point, implying the function is smooth and continuous without sharp corners or breaks.

Chain Rule

A formula used to compute the derivative of a composite function. If y = f(u) and u = g(x), then dy/dx = (dy/du) * (du/dx). It essentially breaks down the differentiation of complex functions into simpler, manageable steps.

Power Rule

A rule for differentiating functions of the form x^n. The derivative of x^n with respect to x is nx^(n-1). This is often used in conjunction with the Chain Rule when differentiating expressions like (f(x))^n.

Step-by-Step Worked Examples

• Example 1: Differentiate y = sin(x^2 + 5) with respect to x Step 1: Identify the outer and inner functions. Let the outer function be f(u) = sin(u) and the inner function be u = g(x) = x^2 + 5. Step 2: Differentiate the outer function with respect to u. dy/du = d/du (sin(u)) = cos(u). Step 3: Differentiate the inner function with respect to x. du/dx = d/dx (x^2 + 5) = 2x. **Step 4: Apply the Chain Rule formula: dy/dx = (dy/du) * (du/dx).** dy/dx = cos(u) (2x). Step 5: Substitute u back with x^2 + 5. dy/dx = cos(x^2 + 5) (2x) = 2x cos(x^2 + 5). Final Answer: dy/dx = 2x cos(x^2 + 5)
• Example 2: Find the derivative of y = (ax + b)^n with respect to x Step 1: Identify outer and inner functions. Let the outer function be f(u) = u^n and the inner function be u = g(x) = ax + b. Step 2: Differentiate the outer function with respect to u. dy/du = d/du (u^n) = n * u^(n-1) (using the Power Rule). Step 3: Differentiate the inner function with respect to x. du/dx = d/dx (ax + b) = a. **Step 4: Apply the Chain Rule formula: dy/dx = (dy/du) * (du/dx).** dy/dx = (n u^(n-1)) a. Step 5: Substitute u back with ax + b. dy/dx = n (ax + b)^(n-1) a = an(ax + b)^(n-1). Final Answer: dy/dx = an(ax + b)^(n-1)
• Example 3: Differentiate y = e^(cos x) with respect to x Step 1: Identify outer and inner functions. Let the outer function be f(u) = e^u and the inner function be u = g(x) = cos x. Step 2: Differentiate the outer function with respect to u. dy/du = d/du (e^u) = e^u. Step 3: Differentiate the inner function with respect to x. du/dx = d/dx (cos x) = -sin x. **Step 4: Apply the Chain Rule formula: dy/dx = (dy/du) * (du/dx).** dy/dx = e^u (-sin x). Step 5: Substitute u back with cos x. dy/dx = e^(cos x) (-sin x) = -sin x e^(cos x). Final Answer: dy/dx = -sin x e^(cos x)

Exam Tip: Avoiding Common Mistakes with the Chain Rule

While the Chain Rule is powerful, students often make a few common errors. Firstly, remember to differentiate all layers of the composite function. A frequent mistake is forgetting to multiply by the derivative of the innermost function. For example, in sin(x^2), d/dx(sin(x^2)) is not just cos(x^2); it's cos(x^2) * d/dx(x^2) = 2x cos(x^2).

Secondly, pay close attention to the order of differentiation. Always start from the outermost function and work inwards. Don't mix up the dy/du and du/dx terms. Finally, be careful with signs, especially when dealing with trigonometric derivatives like d/dx(cos x) = -sin x or d/dx(sec x) = sec x tan x.

Practice Questions with Solutions

• Q: Differentiate y = cos(sin x) with respect to x. A: Step 1: Identify outer function f(u) = cos u and inner function u = sin x. Step 2: Differentiate f(u) w.r.t u: dy/du = -sin u. Step 3: Differentiate u w.r.t x: du/dx = cos x. Step 4: Apply Chain Rule: dy/dx = (dy/du) (du/dx) = (-sin u) (cos x). Step 5: Substitute u = sin x: dy/dx = -sin(sin x) * cos x. Final answer: -cos x sin(sin x)
• Q: Find dy/dx for y = sec(tan(√x)). A: Step 1: Identify layers: y = sec(A), A = tan(B), B = √x = x^(1/2). Step 2: Differentiate each layer: dy/dA = sec A tan A, dA/dB = sec^2 B, dB/dx = (1/2)x^(-1/2) = 1/(2√x). Step 3: Apply Chain Rule: dy/dx = dy/dA dA/dB dB/dx. Step 4: Substitute back: dy/dx = sec(tan(√x)) tan(tan(√x)) sec^2(√x) (1/(2√x)). Final answer: (sec(tan(√x)) tan(tan(√x)) sec^2(√x)) / (2√x)
• Q: Differentiate y = sin((ax+b)/(cx+d)) with respect to x. A: Step 1: Outer function f(u) = sin u, inner function u = (ax+b)/(cx+d). Step 2: dy/du = cos u. Step 3: Differentiate u using Quotient Rule: du/dx = [(cx+d)a - (ax+b)c] / (cx+d)^2 = (acx+ad-acx-bc) / (cx+d)^2 = (ad-bc) / (cx+d)^2. Step 4: Apply Chain Rule: dy/dx = cos u (ad-bc) / (cx+d)^2. Step 5: Substitute u: dy/dx = cos((ax+b)/(cx+d)) (ad-bc) / (cx+d)^2. Final answer: ((ad-bc) / (cx+d)^2) * cos((ax+b)/(cx+d))
• Q: Find the derivative of y = (2x + 3)^5. A: Step 1: Outer function f(u) = u^5, inner function u = 2x + 3. Step 2: Differentiate f(u) w.r.t u: dy/du = 5u^4. Step 3: Differentiate u w.r.t x: du/dx = 2. Step 4: Apply Chain Rule: dy/dx = (5u^4) (2). Step 5: Substitute u = 2x + 3: dy/dx = 5(2x + 3)^4 2 = 10(2x + 3)^4. Final answer: 10(2x + 3)^4

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