Question

3-34 Differentiate the function.

9. \(W\left( v \right) = 1.8{v^{ - 3}}\)

Short answer

The differentiation of the function \(W\left( v \right) = 1.8{v^{ - 3}}\) is \(W'\left( v \right) = - 5.4{v^{ - 4}}\).

Step-by-step solution

Write the formula of the power rule and the derivative of a constant function

The Power Rule:\(\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\) where \(n\) is a positive integer

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 \(W\left( v \right) = 1.8{v^{ - 3}}\). Differentiate the function w.r.t \(v\) by using the power rule and the derivative of a constant function.

\(\begin{aligned}\frac{{d\left( {W\left( v \right)} \right)}}{{dv}} &= \frac{{d\left( {1.8{v^{ - 3}}} \right)}}{{dv}}\\ &= 1.8\frac{d}{{dv}}\left( {{v^{ - 3}}} \right)\\ &= 1.8\frac{d}{{dv}}\left( { - 3{v^{ - 3 - 1}}} \right)\\ &= - 5.4{v^{ - 4}}\end{aligned}\)

Thus, the derivative of the function \(W\left( v \right) = 1.8{v^{ - 3}}\)is \(W'\left( v \right) = - 5.4{v^{ - 4}}\).