Chain Rule Derivative Calculator
Evaluates the derivative of a composite polynomial function y = a(bx + c)ⁿ at a given x value using the chain rule. Ideal for calculus learners verifying step-by-step differentiation of power-of-linear expressions.
About this calculator
The chain rule states: if y = f(g(x)), then dy/dx = f′(g(x)) · g′(x). For y = a(bx + c)ⁿ, the outer function is f(u) = auⁿ and the inner function is g(x) = bx + c. Differentiating step by step: f′(u) = a·n·u^(n−1) and g′(x) = b. Substituting back: dy/dx = a·n·(bx + c)^(n−1)·b. The formula applied here is: derivative = outer_coeff × outer_power × (inner_coeff × x + inner_constant)^(outer_power − 1) × inner_coeff. Evaluating this at a specific x gives the exact slope of the tangent line to the composite function at that point. The key insight is that the inner coefficient b must always appear as a multiplying factor — omitting it is the most common chain rule mistake.
How to use
Differentiate y = 2(4x + 3)³ and evaluate at x = 0. Set outer_coeff = 2, outer_power = 3, inner_coeff = 4, inner_constant = 3, x = 0. Calculation: 2 × 3 × (4×0 + 3)^(3−1) × 4 = 6 × 3² × 4 = 6 × 9 × 4 = 216. So the derivative of y = 2(4x+3)³ at x = 0 is 216, meaning the curve has a slope of 216 at that point.
Frequently asked questions
What is the chain rule and how does it apply to power functions?
The chain rule is a calculus rule for differentiating composite functions — functions built by plugging one function into another. For power functions of the form y = a(g(x))ⁿ, the rule says: multiply the result of the standard power rule by the derivative of the inner function g(x). When g(x) = bx + c (linear), its derivative is simply b, making the chain rule easy to apply. This rule is indispensable in physics, engineering, and economics wherever rates of change involve layered relationships.
How is the chain rule derivative different from the basic power rule derivative?
The basic power rule d/dx[xⁿ] = n·xⁿ⁻¹ only applies when the base is exactly x. The chain rule extends this by accounting for any inner function: d/dx[a(g(x))ⁿ] = a·n·(g(x))^(n−1)·g′(x). The extra factor g′(x) is what sets the chain rule apart. For g(x) = bx + c, g′(x) = b, so the result is a·n·(bx+c)^(n−1)·b. Applying the basic power rule without this multiplier gives the wrong answer whenever the inner function is not simply x.
Why do I need to evaluate the chain rule derivative at a specific x value?
The derivative dy/dx of a non-constant function is itself a function of x, meaning its value changes at every point along the curve. Evaluating at a specific x value converts that general derivative function into a single number — the instantaneous rate of change or tangent slope at exactly that location. This is useful in optimization (finding where the slope equals zero), in physics (computing velocity at a specific time), and in linear approximation (building a tangent line at a given point).