What is the derivative of (t+2t^2)/(1+t^2) ?

The derivative of (t+2t^2)/(1+t^2) is (-t^2+4t+1)/((1+t^2)^2)

What is the first derivative of (t+2t^2)/(1+t^2) ?

The first derivative of (t+2t^2)/(1+t^2) is (-t^2+4t+1)/((1+t^2)^2)

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derivative of

\frac{t + 2 t ^{2}}{1 + t ^{2}}

\frac{- t ^{2} + 4 t + 1}{( 1 + t ^{2} ) ^{2}}

Solution steps

\frac{d}{d t} (\frac{t + 2 t ^{2}}{1 + t ^{2}})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d t} ( t + 2 t ^{2} ) ( 1 + t ^{2} ) - \frac{d}{d t} ( 1 + t ^{2} ) ( t + 2 t ^{2} )}{( 1 + t ^{2} ) ^{2}}

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

\frac{d}{d t} (1 + t^{2}) = 2 t

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

Expand

(1 + 4 t) (1 + t^{2}) - 2 t (t + 2 t^{2}) : - t^{2} + 4 t + 1

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

\int \frac{1}{( e ^{\frac{x}{a}} - e ^{- \frac{x}{a}} ) ^{- 2}} d x

\int e^{x} cos (6 x) d x

\int_{- 1}^{4} 3 x^{2} + 8 x - 5 d x

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

\int \frac{7 - 5 x}{64 - 49 x ^{2}} d x

What is the derivative of (t+2t^2)/(1+t^2) ?
The derivative of (t+2t^2)/(1+t^2) is (-t^2+4t+1)/((1+t^2)^2)
What is the first derivative of (t+2t^2)/(1+t^2) ?
The first derivative of (t+2t^2)/(1+t^2) is (-t^2+4t+1)/((1+t^2)^2)