Question

3-34: Differentiate the function

22. \(S\left( R \right) = 4\pi {R^2}\)

Short answer

The derivative of the function is \(S'\left( R \right) = 8\pi R\).

Step-by-step solution

 Differentiation Rule

When \(c\) is a constant and \(f\) is a differentiable 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 \(n\) is any real number, then;

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

Differentiate the function

Differentiate the function as shown below:

\(\begin{aligned}S'\left( R \right) &= \frac{d}{{dR}}\left( {4\pi {R^2}} \right)\\ &= 4\pi \frac{d}{{dR}}\left( {{R^2}} \right)\\ &= 4\pi \left( 2 \right){R^{2 - 1}}\\ &= 8\pi R\end{aligned}\)

Thus, the derivative of the function is \(S'\left( R \right) = 8\pi R\).