derivative f(x)=5x^2tan(1-sqrt(x))

解答

导数

f (x) = 5 x^{2} tan (1 - x)

5 (2 x tan (1 - x) - \frac{x ^{\frac{3}{2}} sec ^{2} ( - x + 1 )}{2})

求解步骤

\frac{d}{d x} (5 x^{2} tan (1 - x))

将常数提出:

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

= 5 \frac{d}{d x} (x^{2} tan (1 - x))

使用微分乘法定则:

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

f = x^{2}, g = tan (1 - x)

= 5 (\frac{d}{d x} (x^{2}) tan (1 - x) + \frac{d}{d x} (tan (1 - x)) x^{2})

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

\frac{d}{d x} (tan (1 - x)) = - \frac{sec ^{2} ( 1 - x )}{2 x}

= 5 (2 x tan (1 - x) + (- \frac{sec ^{2} ( 1 - x )}{2 x}) x^{2})

化简

5 (2 x tan (1 - x) + (- \frac{sec ^{2} ( 1 - x )}{2 x}) x^{2}) : 5 (2 x tan (1 - x) - \frac{x ^{\frac{3}{2}} sec ^{2} ( - x + 1 )}{2})

= 5 (2 x tan (1 - x) - \frac{x ^{\frac{3}{2}} sec ^{2} ( - x + 1 )}{2})

d er i v a t i v e x (x^{2} + 7)^{4}

t an g e n t 2 x^{3} - 2 x^{2} + 1

x \to 7 + lim (\frac{3}{x - 7})

\int_{5}^{\infty} \frac{1}{x ^{2} - 4} d x

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