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What You'll Learn

• Differentiate any polynomial or power function using the power, constant-multiple, and sum rules — including negative and fractional exponents.
• Rewrite radicals and quotients as powers before differentiating.
• Apply the product and quotient rules, keeping the order and the minus sign correct.
• Recall the derivatives of all six trigonometric functions and evaluate derivatives at a point from a table of values.

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Content

The Power Rule

For any real number n: d/dx[xⁿ] = n·xⁿ⁻¹. Multiply by the old exponent, then subtract 1 from it.

This works for every real exponent — positive, negative, and fractional:

d/dx[x⁷] = 7x⁶
d/dx[x⁻³] = −3x⁻⁴          (−3 − 1 = −4; subtracting 1 moves the exponent DOWN)
d/dx[x^(2/3)] = (2/3)x^(−1/3)

Rewrite first

The power rule needs the form xⁿ. Radicals and reciprocals are disguised powers — undisguise them, differentiate, then (optionally) rewrite back:

d/dx[6√x] = 6·(1/2)x^(−1/2) = 3x^(−1/2) = 3/√x
d/dx[1/x⁴] = d/dx[x⁻⁴] = −4x⁻⁵ = −4/x⁵

Constant, constant-multiple, and sum rules

d/dx[c] = 0                     (a constant is flat)
d/dx[c·f(x)] = c·f'(x)          (coefficient rides along)
d/dx[f ± g] = f' ± g'           (differentiate term by term)

So d/dx[4x³ − 2x² + 7x − 5] = 12x² − 4x + 7 — each term hit with the power rule, and the constant −5 vanishes.

The Product Rule

d/dx[f·g] = f'·g + f·g' — (derivative of the first)(second) + (first)(derivative of the second).

The derivative of a product is not the product of the derivatives. Worked:

d/dx[(x² + 3)(x − 4)] = (2x)(x − 4) + (x² + 3)(1) = 2x² − 8x + x² + 3 = 3x² − 8x + 3

The Quotient Rule

d/dx[f/g] = (f'·g − f·g') / g² (for g ≠ 0).

Two things trip people up: it is a subtraction (opposite of the product rule), and the order matters — f'·g comes first. The chant "Lo·D-Hi − Hi·D-Lo, all over Lo-Lo" keeps the bottom function g (Lo) leading. Simplify the numerator only; leave the denominator as g².

y = (x² − 1)/(x + 2):
y' = [(2x)(x + 2) − (x² − 1)(1)] / (x + 2)²
At x = 1:  [(2)(3) − (0)] / (3)² = 6/9 = 2/3

Simplify first when you can. If the denominator is a single power of x, split the fraction instead of using the quotient rule:

(5x − 2)/x² = 5x⁻¹ − 2x⁻²   →   −5x⁻² + 4x⁻³ = (4 − 5x)/x³

The six trigonometric derivatives

The three "co-" functions (cosine, cotangent, cosecant) are exactly the ones whose derivatives carry a minus sign. Note the common mix-up: tan x gives sec² x, while sec x gives sec x · tan x. All rules assume x is in radians.

Trig derivatives combine with the product and quotient rules:

d/dx[x cos x] = (1)(cos x) + (x)(−sin x) = cos x − x sin x
For g(x) = 5 tan x:  g'(x) = 5 sec² x,  and  g'(π/3) = 5·sec²(π/3) = 5·(2)² = 20

Derivatives at a point from a table

CLEP items often give values of f, g, f', g' at a point and ask for the derivative of a product or quotient there — just substitute:

For h = f·g: h'(3) = f'(3)g(3) + f(3)g'(3) = (5)(−1) + (2)(4) = 3.

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Key Takeaways

• Power rule for every real exponent: d/dx[xⁿ] = n·xⁿ⁻¹; rewrite radicals and reciprocals as powers first.
• Product rule: f'g + fg'. Quotient rule: (f'g − fg')/g² — subtraction, g leads, denominator is g².
• Split a fraction over a single power of x instead of using the quotient rule.
• Memorize the six trig derivatives; the "co-" functions carry the minus sign, and tan → sec², sec → sec·tan.
• For table problems, plug the given values straight into the rule.

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Practice Questions

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Show answer key & explanations

Answer Key

1. E. Correct: term by term, 12x² − 4x + 7, and the constant −5 differentiates to 0. A) keeps −5. B) fails to bring down exponents. C) does not reduce the powers. D) mis-differentiates 7x to 2. Fix rule: every standalone constant differentiates to 0.

2. C. Correct: 1/x⁴ = x⁻⁴, and −4x⁻⁵ = −4/x⁵. A) drops the sign. B) uses exponent −3 (added instead of subtracted). D) inverts the rule. E) drops the coefficient. Fix rule: subtract 1 from −4 to get −5.

3. C. Correct: 6x^(1/2) → 6·(1/2)x^(−1/2) = 3/√x. A) forgets the 1/2. B) adds 1 to the exponent. D) doubles the coefficient. E) drops the coefficient 6. Fix rule: rewrite the radical as a power before differentiating.

4. B. Correct: product rule (2x)(x − 4) + (x² + 3)(1) = 3x² − 8x + 3. A) multiplies the derivatives. C) differentiates only one factor. D) wrong sign on the constant. E) drops the middle term. Fix rule: f'g + fg', never f'g'.

5. C. Correct: y' = [(2x)(x + 2) − (x² − 1)]/(x + 2)²; at x = 1, (6 − 0)/9 = 2/3. A) 6 forgets to square the denominator. B) 1/3 mis-simplifies. D) −2/3 reverses the numerator order. E) 3/2 inverts. Fix rule: Lo·D-Hi first, denominator g².

6. B. Correct: d/dx[tan x] = sec² x. A) is the derivative of sec x. C) is the derivative of cot x. D) is the derivative of cos x. E) is not a derivative here. Fix rule: tan → sec².

7. B. Correct: product rule (1)(cos x) + (x)(−sin x) = cos x − x sin x. A) differentiates only cos x. C) wrong sign (forgets cos differentiates to −sin). D) drops the first term. E) wrong sign on cos x. Fix rule: the "co-" function cosine gives −sin x.

8. D. Correct: g'(x) = 5 sec² x; sec(π/3) = 1/cos(π/3) = 2, so 5·(2)² = 20. A) forgets to square the secant. B) and C) are partial-multiplication slips. E) squares and then doubles. Fix rule: evaluate sec first, then square.

9. D. Correct: h'(3) = f'(3)g(3) + f(3)g'(3) = (5)(−1) + (2)(4) = −5 + 8 = 3. A) multiplies the derivatives. B) keeps only f g'. C) drops the negative on g(3). E) drops the second term. Fix rule: product rule uses both cross terms with correct signs.

10. A. Correct: R = g/f, so R'(2) = [g'(2)f(2) − g(2)f'(2)]/[f(2)]² = [(−3)(4) − (6)(1)]/16 = −18/16 = −9/8. B) reverses the numerator order (wrong sign). C) and D) mishandle the f² denominator. E) wrong sign. Fix rule: the bottom function f sets the f² denominator and leads the numerator with g'·f.

11. E. Correct: split into 5x⁻¹ − 2x⁻², differentiate to −5x⁻² + 4x⁻³ = (4 − 5x)/x³. A) mis-simplifies. B) sign error in the numerator. C) drops a term. D) wrong sign on the leading term. Fix rule: over a single power of x, split the fraction instead of using the quotient rule.

12. A. Correct: d/dx[sec x] = sec x tan x. B) is the derivative of tan x. C) wrong sign — sec x is not a "co-" function. D) is unrelated. E) is the derivative of csc x. Fix rule: sec → sec·tan (positive; only "co-" functions get the minus sign).