Question

3-34 Differentiate the function.

7. \(f\left( t \right) = - 2{e^t}\)

Short answer

The differentiation of the function \(f\left( t \right) = - 2{e^t}\) is \(f'\left( t \right) = - 2{e^t}\).

Step-by-step solution

Write the formula of Derivative of the Natural Exponential Function and the constant multiple rule

Derivative of the Natural Exponential Function: \(\frac{d}{{dx}}\left( {{e^x}} \right) = {e^x}\)

The Constant Multiple Rule: \(\frac{d}{{dx}}\left( {cf\left( x \right)} \right) = c\frac{d}{{dx}}f\left( x \right)\)

Find the differentiation of the function

Consider the function \(f\left( t \right) = - 2{e^t}\). Differentiate the function w.r.t \(t\) by using the derivative of the natural exponential function and the constant multiple rule.

\(\begin{aligned}\frac{{d\left( {f\left( t \right)} \right)}}{{dt}} &= \frac{{d\left( { - 2{e^t}} \right)}}{{dt}}\\ &= \frac{d}{{dt}}\left( { - 2{e^t}} \right)\\ &= - 2\frac{d}{{dt}}\left( {{e^t}} \right)\\ &= - 2{e^t}\end{aligned}\)

Thus, the derivative of the function \(f\left( t \right) = - 2{e^t}\) is \(f'\left( t \right) = - 2{e^t}\).