Lösungen - Differenzieren (Ableiten)

Aufgaben Ableiten

Berechnen Sie die 1. Ableitung.

1.
a) f(x) = x²
f'(x) = 2x

b) f(x) = x⁴
f'(x) = 4x³

c) f(x) = 2x³
f'(x) = 2(3x²) = 6x²

d) f(x) = -3x^(-2)
f'(x) = -3(-2x^(-3)) = 6x^(-3) = 6/x³

e) f(x) = x³ + 5
f'(x) = 3x²

f) f(x) = 3√x
f'(x) = 3/2√x

g) f(x) = 2x⁴ + 3x³
f'(x) = 8x³ + 9x²

2.
a) f(x) = x³sin(x)
f'(x) = 3x²sin(x) + x³cos(x)

b) f(x) = (x³ + 3x²)(x² + 1)
f'(x) = (3x² + 6x)(x² + 1) + (x³ + 3x²)2x
f'(x) = (3x⁴ + 3x² + 6x³ + 6x) + (2x⁴ + 6x³)
f'(x) = 5x⁴ + 12x³ + 3x² + 6x

c) f(x) = e^xln(x)
f'(x) = e^xln(x) + e^x(1/x)

d) f(x) = cos(x)√x
f'(x) = -sin(x)√x + cos(x)[1/(2√x)]
f'(x) = -sin(x)√x + cos(x)/(2√x)

3.
a) f(x) = x⁴/[cos(x)]
f'(x) = [4x³cos(x) - x⁴(-sin(x))]/cos²(x)

b) f(x) = e^x/x³
f'(x) = [e^xx³ - e^x3x²]/x^(3·2)
f'(x) = [e^x(x³ - 3x²)]/x⁶

c) f(x) = x/[ln(x)]
f'(x) = [1ln(x) - x(1/x)]/[(ln(x))²]
f'(x) = [ln(x) - 1]/[(ln(x))²]

d) f(x) = tan(x) = sin(x)/cos(x)
f'(x) = [cos(x)cos(x) - sin(x)(-sin(x)]/[cos²(x)]
f'(x) = [cos²(x) + sin²(x)]/[cos²(x)]
(cos²(x) + sin²(x) = 1 ; siehe Additionstheoreme)
f'(x) = 1/[cos²(x)]

4.
a) f(x) = cos(x⁴)
f'(x) = -sin(x⁴)4x³

b) f(x) = e^(x³)
f'(x) = e^(x³)3x²

c) f(x) = √[1+sin(x³)]
f'(x) = [1/(2√[1+sin(x³)]]cos(x³)3x²

d) f(x) = [1+ ln(x⁴)]²
f'(x) = 2[1+ ln(x⁴)](1/x⁴)4x³

e) f(x) = (3x)^(1-2x) = e^{ln(3x^(1-2x))} = e^(1-2x)ln(3x)
f'(x) = e^(1-2x)ln(3x)[-2ln(3x) + (1-2x)(3/3x)]

Die Aufgaben zu diesen Lösungen finden Sie hier.