Question

3-34: Differentiate the function.

18. \(W\left( t \right) = \sqrt t - 2{e^t}\)

Short answer

The derivative of the function is \(W'\left( t \right) = \frac{1}{{2\sqrt t }} - 2{e^t}\).

Step-by-step solution

 Differentiation Rule

When \(c\) is a constant and \(f\) is adifferentiable function, then;

\(\frac{d}{{dx}}\left( {cf\left( x \right)} \right) = c\frac{d}{{dx}}f\left( x \right)\)

When \(n\) is any real number, then;

\(\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{x - 1}}\)

The derivative of the exponential function is shown below:

\(\frac{d}{{dx}}\left( {{e^x}} \right) = {e^x}\)

Differentiate the function

Rewrite the function as the power of \(t\) as shown below:

\(W\left( t \right) = {t^{\frac{1}{2}}} - 2{e^t}\)

Differentiate the function as shown below:

\(\begin{aligned}W'\left( t \right) &= \frac{d}{{dt}}\left( {{t^{\frac{1}{2}}} - 2{e^t}} \right)\\ &= \frac{d}{{dt}}\left( {{t^{\frac{1}{2}}}} \right) - 2\frac{d}{{dt}}\left( {{e^t}} \right)\\ &= \frac{1}{2}{t^{\frac{1}{2} - 1}} - 2\left( {{e^t}} \right)\\ &= \frac{1}{2}{t^{ - \frac{1}{2}}} - 2{e^t}\\ &= \frac{1}{{2\sqrt t }} - 2{e^t}\end{aligned}\)

Thus, the derivative of the function is \(W'\left( t \right) = \frac{1}{{2\sqrt t }} - 2{e^t}\).