Question

3-34: Differentiate the function.

16. \(h\left( w \right) = \sqrt 2 w - \sqrt 2 \)

Short answer

The derivative of the function is \(\sqrt 2 \).

Step-by-step solution

Differentiation Rule

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

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

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

The derivative of the constant function is;

\(\frac{d}{{dx}}\left( c \right) = 0\)

When \(f\) and \(g\) are both differentiable, then;

\(\begin{aligned}\frac{d}{{dx}}\left( {f\left( x \right) + g\left( x \right)} \right) &= \frac{d}{{dx}}f\left( x \right) + \frac{d}{{dx}}g\left( x \right)\\\frac{d}{{dx}}\left( {f\left( x \right) - g\left( x \right)} \right) &= \frac{d}{{dx}}f\left( x \right) - \frac{d}{{dx}}g\left( x \right)\end{aligned}\)

Differentiate the function

Differentiate the function as shown below:

\(\begin{aligned}h'\left( w \right) &= \frac{d}{{dw}}\left( {\sqrt 2 w - \sqrt 2 } \right)\\ &= \sqrt 2 \frac{d}{{dw}}\left( w \right) - \frac{d}{{dw}}\left( {\sqrt 2 } \right)\\ &= \sqrt 2 \left( 1 \right) - 0\\ &= \sqrt 2 \end{aligned}\)

Thus, the derivative of the function is \(h'\left( w \right) = \sqrt 2 \).