Let’s solve each problem one by one. We’re finding the derivative of each function at x = p, which means we’ll first find the general derivative f’(x), then plug in p for x.
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Problem 1:
f(x) = -6x² - 9x + 7
Derivative rule: d/dx [axⁿ] = n·a·x^(n-1)
So:
f’(x) = 2·(-6)x¹ + 1·(-9)x⁰ + 0
f’(x) = -12x - 9
Then f’(p) = -12p - 9
✔ Check: Power rule applied correctly. Constants disappear. Looks good.
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Problem 2:
f(x) = -5x² - 9x - 2
f’(x) = 2·(-5)x + (-9) = -10x - 9
f’(p) = -10p - 9
✔ Same as above — correct.
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Problem 3:
f(x) = x² - 7x - 9
f’(x) = 2x - 7
f’(p) = 2p - 7
✔ Simple and correct.
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Problem 4:
f(x) = (3x + 3)/(5x - 3) → quotient rule needed!
Quotient rule: if f(x) = u/v, then f’(x) = (v·u’ - u·v’) / v²
Here:
u = 3x + 3 → u’ = 3
v = 5x - 3 → v’ = 5
f’(x) = [(5x - 3)(3) - (3x + 3)(5)] / (5x - 3)²
= [15x - 9 - (15x + 15)] / (5x - 3)²
= [15x - 9 - 15x - 15] / (5x - 3)²
= (-24) / (5x - 3)²
So f’(p) = -24 / (5p - 3)²
✔ Verified algebra — signs and distribution correct.
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Problem 5:
f(x) = (-3x + 4)/(3x - 10)
Again, quotient rule:
u = -3x + 4 → u’ = -3
v = 3x - 10 → v’ = 3
f’(x) = [ (3x - 10)(-3) - (-3x + 4)(3) ] / (3x - 10)²
= [ -9x + 30 - (-9x + 12) ] / (3x - 10)²
= [ -9x + 30 + 9x - 12 ] / (3x - 10)²
= 18 / (3x - 10)²
So f’(p) = 18 / (3p - 10)²
✔ Correct — negative signs handled properly.
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Problem 6:
f(x) = (x - 3)/(8x - 7)
Quotient rule again:
u = x - 3 → u’ = 1
v = 8x - 7 → v’ = 8
f’(x) = [ (8x - 7)(1) - (x - 3)(8) ] / (8x - 7)²
= [8x - 7 - (8x - 24)] / (8x - 7)²
= [8x - 7 - 8x + 24] / (8x - 7)²
= 17 / (8x - 7)²
So f’(p) = 17 / (8p - 7)²
✔ Good — constants combined right.
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Problem 7:
f(x) = 5√x = 5x^(1/2)
Power rule: d/dx [x^n] = n·x^(n-1)
f’(x) = 5 · (1/2) x^(-1/2) = (5/2) / √x
So f’(p) = 5/(2√p)
✔ Correct — fractional exponent handled properly.
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Problem 8:
f(x) = -13√x = -13x^(1/2)
f’(x) = -13 · (1/2) x^(-1/2) = -13/(2√x)
So f’(p) = -13/(2√p)
✔ Same logic as #7 — sign preserved.
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Final Answers:
1. -12p - 9
2. -10p - 9
3. 2p - 7
4. -24 / (5p - 3)²
5. 18 / (3p - 10)²
6. 17 / (8p - 7)²
7. 5/(2√p)
8. -13/(2√p)
──────────────────────────────────────
Final Answer:
1. $ f'(p) = -12p - 9 $
2. $ f'(p) = -10p - 9 $
3. $ f'(p) = 2p - 7 $
4. $ f'(p) = \frac{-24}{(5p - 3)^2} $
5. $ f'(p) = \frac{18}{(3p - 10)^2} $
6. $ f'(p) = \frac{17}{(8p - 7)^2} $
7. $ f'(p) = \frac{5}{2\sqrt{p}} $
8. $ f'(p) = \frac{-13}{2\sqrt{p}} $
Parent Tip: Review the logic above to help your child master the concept of free precalculus worksheet.
