derivative of (w(x,y(w,x))(x^2+y(w,x)^2))

解答

\frac{d}{d x} ((w (x, y (w, x))) (x^{2} + y (w, x) (w, x)^{2}))

w ((, (1 \cdot y (w, x) + \frac{d}{d x} (y (w, x)) x)) (x^{2} + y (w, x) (w, x)^{2}) + (2 x + (\frac{d}{d x} (y (w, x)) (w, x)^{2} + 2 (w, x) (w, 1) y (w, x))) (x, y (w, x)))

求解步骤

\frac{d}{d x} ((w (x, y (w, x))) (x^{2} + y (w, x) (w, x)^{2}))

将

w

视为常数

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= w \frac{d}{d x} ((x, y (w, x)) (x^{2} + y (w, x) (w, x)^{2}))

使用微分乘法定则:

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

f = x, y (w, x), g = x^{2} + y (w, x) (w, x)^{2}

= w (\frac{d}{d x} (x, y (w, x)) (x^{2} + y (w, x) (w, x)^{2}) + \frac{d}{d x} (x^{2} + y (w, x) (w, x)^{2}) (x, y (w, x)))

\frac{d}{d x} (x, y (w, x)) = (, (1 \cdot y (w, x) + \frac{d}{d x} (y (w, x)) x))

\frac{d}{d x} (x^{2} + y (w, x) (w, x)^{2}) = 2 x + (\frac{d}{d x} (y (w, x)) (w, x)^{2} + 2 (w, x) (w, 1) y (w, x))

= w ((, (1 \cdot y (w, x) + \frac{d}{d x} (y (w, x)) x)) (x^{2} + y (w, x) (w, x)^{2}) + (2 x + (\frac{d}{d x} (y (w, x)) (w, x)^{2} + 2 (w, x) (w, 1) y (w, x))) (x, y (w, x)))

e x p an d \frac{1}{( x + 3 ) ^{5}}

e x p an d (y^{''} (t) + y)^{2}

e x p an d \frac{2 x + 7}{( x + 3 ) ( x + 4 )}

e x p an d 16 (3 i + 1)^{8}

e x p an d y (y d x - x d y) + 3 y^{4} - x^{4} (y d x + x d y)