Example 1 · level 1
A cost function is C(x) = 5x² + 5x − 4. Find the marginal cost C′(x) at x = 1.
Show solution
- C′(x) = 10x + 5.
- C′(1) = 10(1) + (5) = 15.
Answer: 15
CA FOUNDATION: QUANTITATIVE APTITUDE · BASIC CALCULUS
Differentiation: derivative of a polynomial at a point
d/dx (xⁿ) = n·xⁿ⁻¹; constants differentiate to 0. Differentiate term by term, then substitute.
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Worked examples
Example 2 · level 2
A cost function is C(x) = −5x³ − 4x² + 8x − 7. Find the marginal cost C′(x) at x = 3.
Show solution
- C′(x) = −15x² − 8x + 8.
- C′(3) = -15(3)² + -8(3) + (8) = -151.
Answer: -151
Example 3 · level 3
A cost function is C(x) = 3x³ − 5x² − 2x + 4. Find the marginal cost C′(x) at x = 2.
Show solution
- C′(x) = 9x² − 10x − 2.
- C′(2) = 9(2)² + -10(2) + (-2) = 14.
Answer: 14