- Problem 2: f'(x) = ln(x)
- Problem 3: f'(x) = cos(ln(x)) / x
- Problem 4: f'(x) = 2 cot(x)
- Problem 5: f'(x) = -1/x
- Problem 6: y' = -1 / (x (ln x)^2)
- Problem 7: f'(x) = -sin(x) / ((1 + cos(x)) ln(10))
- Problem 8: f'(x) = 1 / (2x ln(10))
- Problem 9: g'(x) = (1 - 2x) / x
- Problem 10: g'(t) = 1 / (2√(1 + ln t) * t)
- Problem 11: F'(t) = (2 sin(t) ln(t) + (ln(t))^2 cos(t)) / t
- Problem 12: h'(x) = 1 / √(x^2 - 1)
- Problem 13: G'(y) = (2(2y+1)(y^2+1) - (2y+1)^5) / ((2y+1)^4 √(y^2+1))
- Problem 14: P'(v) = (1 - v + v ln(v)) / (v (1 - v)^2)
- Problem 15: F'(s) = 1 / (s ln(s))
- Problem 16: y' = (1 - 3t^2) / (1 + t - t^3)
- Problem 17: T'(z) = 2^z ln(2) log_2(z) + 2^z / (z ln(2))
- Problem 18: y' = (csc(x) - cot(x))' / (csc(x) - cot(x)) = (-csc(x)cot(x) + csc^2(x)) / (csc(x) - cot(x)) = csc(x)
- Problem 19: y' = -1 - x / (e^(-x) + x e^(-x))
- Problem 20: H'(z) = -z / (a^2 - z^2)
- Problem 21: y' = sec^2(ln(ax+b)) * a / (ax+b)
- Problem 22: y' = (log_5(x) + x log_5(x) * (1/(x ln(5)))) / (x log_5(x) ln(2))
- Problem 23: y' = (1 + 2 ln(x)) / (2√x), y'' = (1 - 2 ln(x)) / (4x√x)
- Problem 24: y' = 1 / (x (1 + ln(x))^2), y'' = -(2 + ln(x)) / (x^2 (1 + ln(x))^3)
- Problem 25: y' = tan(x)
- Problem 26: y' = 1 / (x (1 + ln(x))), y'' = -(1 + ln(x) + 1) / (x^2 (1 + ln(x))^2) = -(2 + ln(x)) / (x^2 (1 + ln(x))^2)
- Problem 27: f'(x) = (1 - ln(x-1) + 1) / (1 - ln(x-1))^2 = (2 - ln(x-1)) / (1 - ln(x-1))^2, Domain: x > 1 and x ≠ 1 + e
- Problem 28: f'(x) = 1 / (2(x + ln(x))^(1/2) * (1 + 1/x)), Domain: x > 0 and x + ln(x) ≥ 0
- Problem 29: f'(x) = (2x - 2) / (x^2 - 2x), Domain: x < 0 or x > 2
- Problem 30: f'(x) = 1 / (x ln(x) ln(ln(x))), Domain: x > 1
- Problem 31: f'(1) = 2
- Problem 32: f'(1) = -2 sin(ln(1)) / 1 = 0
- Problem 33: y = (3/7)(x - 3)
- Problem 34: y = x - 1
- Problem 35: f'(x) = cos(x) + 1/x
- Problem 36: At (1,0): y = x - 1; At (e, 1/e): y = (1/e^2)(x - e) + 1/e
- Problem 37: c = 6
- Problem 38: b = e^3
- Problem 39: y' = 2(x^2+2)(x^4+4)^4 * (2x(x^4+4) + 4x^3(x^2+2))
- Problem 40: y' = [(-e^(-x) cos^2(x) (x^2+x+1) - e^(-x) (2 cos(x)(-sin(x))(x^2+x+1) - e^(-x) cos^2(x) (2x+1))] / (x^2+x+1)^2
- Problem 41: y' = [(1/2)√((x-1)/(x^4+1)) * ((x^4+1) - (x-1)(4x^3)) / (x^4+1)^2]
- Problem 42: y' = e^(x^2-x) * [ (1/(2√x)) + √x (2x-1) + (2/3)(x+1)^(-1/3) ]
- Problem 43: y' = x^x (1 + ln(x))
- Problem 44: y' = x^(cos(x)) * (cos(x)/x - sin(x) ln(x))
- Problem 45: y' = x^(sin(x)) * (sin(x)/x + cos(x) ln(x))
- Problem 46: y' = √x^x * (1/2 + (1/2) ln(x))
- Problem 47: y' = (cos(x))^x * (ln(cos(x)) - x tan(x))
- Problem 48: y' = (sin(x))^(ln(x)) * (ln(sin(x))/x + ln(x) cot(x))
- Problem 49: y' = (tan(x))^(1/x) * (sec^2(x)/(x tan(x)) - ln(tan(x))/x^2)
- Problem 50: y' = (ln(x))^(cos(x)) * (cos(x)/(x ln(x)) - sin(x) ln(ln(x)))
- Problem 51: y' = 2x / (x^2 + y^2)
- Problem 52: y' = y^2 / (x y - x^2 ln(y))
- Problem 53: f^(n)(x) = (-1)^(n-1) (n-1)! / (x-1)^n
- Problem 54: d^9/dx^9 (x^8 ln(x)) = 8! / x
- Problem 55: The limit is the definition of the derivative of ln(1+x) at x=0, which equals 1.
- Problem 56: Let L = lim_{n→∞} (1 + x/n)^n. Then ln(L) = lim_{n→∞} n ln(1 + x/n) = lim_{n→∞} (ln(1 + x/n)) / (1/n). By L'Hôpital's Rule, this equals lim_{n→∞} (1/(1+x/n)) * (-x/n^2) / (-1/n^2) = x. Thus, L = e^x.
Parent Tip: Review the logic above to help your child master the concept of derivatives of logarithmic functions worksheet.
