derivative of (x^{-2}+x^3(4x)^{-2x})

Lösung

\frac{d}{d x} ((x^{- 2} + x)^{3} (4 x)^{- 2 x})

3 \cdot 1 6^{- x} x^{- 2 x} (\frac{1}{x ^{2}} + x)^{2} (- \frac{2}{x ^{3}} + 1) + 1 6^{- x} x^{- 2 x} (- 2 ln (4 x) - 2) (\frac{1}{x ^{2}} + x)^{3}

Schritte zur Lösung

\frac{d}{d x} ((x^{- 2} + x)^{3} (4 x)^{- 2 x})

Wende die Produktregel an:

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

f = (x^{- 2} + x)^{3}, g = (4 x)^{- 2 x}

= \frac{d}{d x} ((x^{- 2} + x)^{3}) (4 x)^{- 2 x} + \frac{d}{d x} ((4 x)^{- 2 x}) (x^{- 2} + x)^{3}

\frac{d}{d x} ((x^{- 2} + x)^{3}) = 3 (\frac{1}{x ^{2}} + x)^{2} (- \frac{2}{x ^{3}} + 1)

\frac{d}{d x} ((4 x)^{- 2 x}) = 1 6^{- x} x^{- 2 x} (- 2 ln (4 x) - 2)

= 3 (\frac{1}{x ^{2}} + x)^{2} (- \frac{2}{x ^{3}} + 1) (4 x)^{- 2 x} + 1 6^{- x} x^{- 2 x} (- 2 ln (4 x) - 2) (x^{- 2} + x)^{3}

Vereinfache

3 (\frac{1}{x ^{2}} + x)^{2} (- \frac{2}{x ^{3}} + 1) (4 x)^{- 2 x} + 1 6^{- x} x^{- 2 x} (- 2 ln (4 x) - 2) (x^{- 2} + x)^{3} : 3 \cdot 1 6^{- x} x^{- 2 x} (\frac{1}{x ^{2}} + x)^{2} (- \frac{2}{x ^{3}} + 1) + 1 6^{- x} x^{- 2 x} (- 2 ln (4 x) - 2) (\frac{1}{x ^{2}} + x)^{3}

= 3 \cdot 1 6^{- x} x^{- 2 x} (\frac{1}{x ^{2}} + x)^{2} (- \frac{2}{x ^{3}} + 1) + 1 6^{- x} x^{- 2 x} (- 2 ln (4 x) - 2) (\frac{1}{x ^{2}} + x)^{3}

x y^{^{'}} = 4 x^{4} cos (2 x) + 3 y

x \to 0 lim (x sin (\frac{π}{x}))

a re a y = 5 x - x^{2}, y = 2 x, (3, 6)

\frac{\partial}{\partial x} (4 y - 3 x^{2})

t an g e n t f (x) = x^{3} - 3 x + 2 at (2.4)