Berechnen Sie die Ableitungen der Funktionen: f(x) = sin(e^x), f(x) = 1/√x, f(x) = x · ln(x), f(x) = x^x

Question

Berechnen Sie die Ableitungen folgender Funktionen:

f(x) = (x^{2} - 1)^{3}

f(x) = sin(e^{x})

f(x) = 1/√x

f(x) = x · ln(x)

f(x) = arccos(x)

f(x) = e^{x} · (x^{4} - 2x)

f(x) = cos(x) / sin(x)

f(x) = x^{x}

Answer 1

f(x) = (x^2 - 1)^3
f '(x) = 3(x^2 - 1)^2 * 2x = 6x * (x^2 - 1)^2

f(x) = sin(e^x)
f '(x) = cos(e^x) * e^x

f(x) = 1/√x = x^{-1/2}
f '(x) = -1/2*x^{-3/2}

f(x) = x * ln(x)
f '(x) = 1 * ln(x) + x * 1/x = ln(x) + 1

f(x) = arccos(x)
f '(x) = - 1/√(1 - x^2)

f(x) = e^x * (x^4 - 2x)
f '(x) = e^x * (x^4 - 2x) + e^x * (4x^3 - 2) = e^x * (x^4 + 4x^3 - 2x - 2)

f(x) = cos(x) / sin(x)
f '(x) = (-sin(x) * sin(x) - cos(x) * cos(x)) / sin^2(x) = -1 / sin^2(x)

f(x) = x^x = e^{x * ln(x)}
f '(x) = e^{x * ln(x)} * (ln(x) + 1) = x^x * (ln(x) + 1)