Chain Rule Examples

Business Calculus · Axiom Academy

Differentiating composite functions step-by-step "The derivative of the outside times the derivative of the inside" How to Identify Composite Functions Look for a function "inside" another function. If you can describe it as "f of something," where "something" isn't just x, you need the chain rule. Find the derivative of f(x) = (3x + 5)^4 Example 2: Exponential with Linear Inside Find the derivative of g(x) = e^(5x-2) Example 3: Natural Log with Quadratic Inside Find the derivative of h(x) = ln(x^2 + 1) Example 4: Square Root (Fractional Exponent) Find the derivative of k(x) = sqrt(4x^2 - x) A company's revenue depends on price p, and price depends on time t: Find the rate of change of revenue with respect to time. At any time t, revenue is changing at a rate of -20(50 - 2t) = -1000 + 40t dollars per month. Initially (t=0), revenue decreases by 1000/month, but this rate of decrease slows over time. Based on the example above, which approach is correct?

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