derivative y=(k(x)^2+x^2)^4(x+k(x))

解答

导数

y = (k (x)^{2} + x^{2})^{4} (x + k (x))

4 (k x^{2} + x^{2})^{3} (2 k x + 2 x) (x + k x) + (1 + k) (k x^{2} + x^{2})^{4}

求解步骤

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

使用微分乘法定则:

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

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

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

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

\frac{d}{d x} (x + k x) = 1 + k

= 4 (k x^{2} + x^{2})^{3} (2 k x + 2 x) (x + k x) + (1 + k) (k x^{2} + x^{2})^{4}

\frac{\partial}{\partial x} (x^{2} y + y ln (x z^{3}))

\int 3 x^{4} + \frac{7}{4} x^{5} d x

d er i v a t i v e y = 2 ln (3) + \frac{1}{2} ln (x^{2} + 1)

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

\int se x^{2} x d x