91Ó°ÊÓ

When differentiating functions, it's important to understand the constant multiple rule. This rule states that if you have a constant multiplied by a function, the derivative of the entire expression can be found by multiplying the constant by the derivative of the function.
If your function is in the form of \( k \times f(x) \), where \( k \) is a constant, then the derivative \( \frac{d}{dx}[k \times f(x)] \) is simply \( k \times \frac{d}{dx}[f(x)] \).
In the given exercise, the function \( v(r)=\frac{4}{3} \times \pi \times r^{3} \) has a constant multiplier \( \frac{4}{3} \pi \). Applying the constant multiple rule allows us to write the derivative as:
\( \frac{d}{dr}[v(r)] = \frac{4}{3} \pi \times \frac{d}{dr} [r^{3}] \).
This simplifies the problem, allowing us to focus only on differentiating the function \( r^{3} \).

One of the most useful rules in differentiation is the power rule. The power rule is essential when differentiating functions of the form \( x^{n} \), where \( n \) is any real number.
According to the power rule, the derivative of \( x^{n} \) with respect to \( x \) is \( n \times x^{n-1} \).
In the exercise, we need to differentiate \( r^{3} \). Applying the power rule, we get:
\( \frac{d}{dr}[r^{3}] = 3r^{2} \).
This shows how the power rule helps to quickly find the derivative of polynomial functions, saving us from more complex operations. It's especially important for students to get comfortable with this rule, as it's used very frequently in calculus.

The concept of a derivative is fundamental in calculus. It measures how a function changes as its input changes, essentially capturing the rate of change or the slope of the function.
For a function \( f(x) \), the derivative \( f'(x) \) is described as the limit:
\lim_{{h \rightarrow 0}} \frac{{f(x+h) - f(x)}}{h}
This formal definition is crucial but in practice, using differentiation rules like the constant multiple rule or power rule simplifies finding derivatives.
In our given function \( v(r)=\frac{4}{3} \pi r^{3} \), we found that:
\( \frac{d}{dr} [v(r)] = \frac{4}{3} \pi \times \frac{d}{dr} [r^{3}] \) Applying the power rule, we determined:
\( \frac{d}{dr} [r^{3}] = 3r^{2} \). Finally, combining the results gives us the derivative:
\( \frac{d}{dr} [v(r)] = 4 \pi r^{2} \).
The clear understanding of derivatives is crucial for further studies in calculus, such as integration, optimization, and differential equations.