Question

For a positive constant \( a \), differentiate \( \left( t + \frac{1}{t} \right)^a \) with respect to \( t \), where \( t \) is a non-zero real number.

Hint:

For differentiating composite functions, use the chain rule. In cases like these, also remember to differentiate the inner function \( t + \frac{1}{t} \).

Answer 1

The function \( f(t) = \left( t + \frac{1}{t} \right)^a \) is to be differentiated with respect to \( t \). Applying the chain rule: \[ \frac{d}{dt} \left( t + \frac{1}{t} \right)^a = a \left( t + \frac{1}{t} \right)^{a-1} \cdot \frac{d}{dt} \left( t + \frac{1}{t} \right) \] The derivative of \( t + \frac{1}{t} \) is: \[ \frac{d}{dt} \left( t + \frac{1}{t} \right) = 1 - \frac{1}{t^2} \] Consequently, the derivative is: \[ \frac{d}{dt} \left( t + \frac{1}{t} \right)^a = a \left( t + \frac{1}{t} \right)^{a-1} \left( 1 - \frac{1}{t^2} \right) \]