derivative tan(x-y)= y/(1+x^2)

解答

导数

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

\frac{d y}{d x} = - \frac{- 2 x y - sec ^{2} ( x - y ) - 2 x ^{2} sec ^{2} ( x - y ) - x ^{4} sec ^{2} ( x - y )}{sec ^{2} ( x - y ) + x ^{2} + 1 + x ^{4} sec ^{2} ( x - y ) + 2 x ^{2} sec ^{2} ( x - y )}

求解步骤

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

将

y

视为

y (x)

对两侧微分:

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

整理

\frac{d y}{d x} : \frac{d y}{d x} = - \frac{- 2 x y - sec ^{2} ( x - y ) - 2 x ^{2} sec ^{2} ( x - y ) - x ^{4} sec ^{2} ( x - y )}{sec ^{2} ( x - y ) + x ^{2} + 1 + x ^{4} sec ^{2} ( x - y ) + 2 x ^{2} sec ^{2} ( x - y )}

\frac{d y}{d x} = - \frac{- 2 x y - sec ^{2} ( x - y ) - 2 x ^{2} sec ^{2} ( x - y ) - x ^{4} sec ^{2} ( x - y )}{sec ^{2} ( x - y ) + x ^{2} + 1 + x ^{4} sec ^{2} ( x - y ) + 2 x ^{2} sec ^{2} ( x - y )}

t an g e n t e^{x}

d er i v a t i v e f (x) = e^{x}

y = 8 x^{2}

d er i v a t i v e x^{2} + x + 1

d er i v a t i v e y = 1 - x^{2}