derivative of x^msqrt(a^2-x^2)

Lösung

\frac{d}{d x} (x^{m} a^{2} - x^{2})

m x^{m - 1} a^{2} - x^{2} - \frac{x ^{1 + m}}{a ^{2} - x ^{2}}

Schritte zur Lösung

\frac{d}{d x} (x^{m} a^{2} - x^{2})

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{m}, g = a^{2} - x^{2}

= \frac{d}{d x} (x^{m}) a^{2} - x^{2} + \frac{d}{d x} (a^{2} - x^{2}) x^{m}

\frac{d}{d x} (x^{m}) = m x^{m - 1}

\frac{d}{d x} (a^{2} - x^{2}) = - \frac{x}{a ^{2} - x ^{2}}

= m x^{m - 1} a^{2} - x^{2} + (- \frac{x}{a ^{2} - x ^{2}}) x^{m}

Vereinfache

m x^{m - 1} a^{2} - x^{2} + (- \frac{x}{a ^{2} - x ^{2}}) x^{m} : m x^{m - 1} a^{2} - x^{2} - \frac{x ^{1 + m}}{a ^{2} - x ^{2}}

= m x^{m - 1} a^{2} - x^{2} - \frac{x ^{1 + m}}{a ^{2} - x ^{2}}

\frac{\partial}{\partial x} (x^{2} + 8 x y + 17 y^{2} - 2 y + 1)

\frac{\partial}{\partial x} (sin (2 t))

\int (- 8 x^{7} y^{- 5} + 6 x^{6} y^{- 4}) d y

\int_{1}^{t} \frac{31}{x ^{3}} d x

x \to 0 lim (\frac{ln ^{2} ( x )}{x})