derivative (4t)/(t^2+25)

解答

导数

\frac{4 t}{t ^{2} + 25}

\frac{4 ( - t ^{2} + 25 )}{( t ^{2} + 25 ) ^{2}}

求解步骤

\frac{d}{d t} (\frac{4 t}{t ^{2} + 25})

将常数提出:

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

= 4 \frac{d}{d t} (\frac{t}{t ^{2} + 25})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= 4 \cdot \frac{\frac{d t}{d t} ( t ^{2} + 25 ) - \frac{d}{d t} ( t ^{2} + 25 ) t}{( t ^{2} + 25 ) ^{2}}

\frac{d t}{d t} = 1

\frac{d}{d t} (t^{2} + 25) = 2 t

= 4 \cdot \frac{1 \cdot ( t ^{2} + 25 ) - 2 tt}{( t ^{2} + 25 ) ^{2}}

化简

4 \cdot \frac{1 \cdot ( t ^{2} + 25 ) - 2 tt}{( t ^{2} + 25 ) ^{2}} : \frac{4 ( - t ^{2} + 25 )}{( t ^{2} + 25 ) ^{2}}

= \frac{4 ( - t ^{2} + 25 )}{( t ^{2} + 25 ) ^{2}}

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