Question 1

When do I use the chain rule?

Accepted Answer

Use the chain rule whenever a function is composed inside another — whenever you can identify an "inner" and "outer" function. Signs you need it: the variable appears inside a power, inside trig/exp/log, or inside any non-trivial expression. Examples: (2x+1)⁵, sin(x²), ln(3x−1), e^(cos x). If the inner function is just x (like sin(x)), no chain rule is needed.

Question 2

How do I identify the inner and outer functions?

Accepted Answer

Ask: what would I compute last if evaluating at a specific x? That's the outer function. Everything inside it is the inner function. In √(x²+1), you'd first compute x²+1, then take the square root — so outer = √u, inner = x²+1. In sin²(x) = (sin x)², outer = u², inner = sin x.

Question 3

What is the chain rule with the power rule?

Accepted Answer

The generalized power rule combines both: d/dx[(g(x))ⁿ] = n·(g(x))ⁿ⁻¹ · g'(x). Example: d/dx[(3x²+1)⁴] = 4(3x²+1)³ · 6x = 24x(3x²+1)³. This is the most frequently used chain rule application in introductory calculus.

Question 4

How do I apply the chain rule to exponential functions?

Accepted Answer

d/dx[eᵍ⁽ˣ⁾] = eᵍ⁽ˣ⁾ · g'(x). Example: d/dx[e^(5x³)] = e^(5x³) · 15x² = 15x²e^(5x³). For other bases: d/dx[aᵍ⁽ˣ⁾] = aᵍ⁽ˣ⁾ · ln(a) · g'(x).

Question 5

How do I apply the chain rule to logarithmic functions?

Accepted Answer

d/dx[ln(g(x))] = g'(x)/g(x). Example: d/dx[ln(x²+5)] = 2x/(x²+5). More generally, d/dx[logₐ(g(x))] = g'(x)/(g(x)·ln a). This appears often in logarithmic differentiation for complicated products.

Question 6

Can the chain rule apply multiple times?

Accepted Answer

Yes — for compositions of three or more functions, apply the chain rule repeatedly from outside in. For y = sin(e^(x²)): first d/dx[sin(u)] = cos(u), then d/dx[eᵛ] = eᵛ, then d/dx[x²] = 2x. Result: cos(e^(x²)) · e^(x²) · 2x = 2x·e^(x²)·cos(e^(x²)).

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