Problem 25 - Calculus for Biology and Medicine

Question

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ g(s)=\left(3 s^{7}-7 s\right)^{3 / 2} $$

Step-by-Step Solution

Answer

The derivative of the function is \( \frac{3}{2}(3s^7 - 7s)^{1/2} (21s^6 - 7) \).

Step 1: Identify the Outer and Inner Functions

The given function is \( g(s) = (3s^7 - 7s)^{3/2} \). It is in the form of a composite function, where the outer function is \( u^{3/2} \) and the inner function is \( u = 3s^7 - 7s \).

Step 2: Differentiate the Outer Function

Differentiate the outer function \( u^{3/2} \) with respect to \( u \). The derivative is \( \frac{d}{du}(u^{3/2}) = \frac{3}{2}u^{1/2} \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( 3s^7 - 7s \) with respect to \( s \). The derivative is \( \frac{d}{ds}(3s^7 - 7s) = 21s^6 - 7 \).

Step 4: Apply the Chain Rule

To find the derivative of \( g(s) \) with respect to \( s \), use the chain rule: \( g'(s) = \frac{d}{du}(u^{3/2}) \cdot \frac{d}{ds}(3s^7 - 7s) \). This gives \( g'(s) = \frac{3}{2}(3s^7 - 7s)^{1/2} \cdot (21s^6 - 7) \).

Step 5: Simplify the Expression

Combine the results to obtain the simplified expression for the derivative: \( g'(s) = \frac{3}{2}(3s^7 - 7s)^{1/2} (21s^6 - 7) \).

Key Concepts

Composite Function

When differentiating a function like the one given in the exercise, it is important to recognize the presence of a composite function. A composite function is a function made up of an "outer" function and an "inner" function. In simpler terms, it is when one function is plugged into another function.

In the problem, the function \( g(s) = (3s^7 - 7s)^{3/2} \) involves a composition. Here, the inner function is \( 3s^7 - 7s \) and it gets enclosed in an outer function, which is raising it to the power \( \frac{3}{2} \). Recognizing this structure is crucial because it tells us we're dealing with a situation where the chain rule will be useful. Understanding composite functions helps us break down complex problems into manageable parts, making differentiation simpler.

Chain Rule

The chain rule is an essential tool in calculus, especially for finding the derivative of composite functions. This rule states that to differentiate a composite function, you multiply the derivative of the outer function by the derivative of the inner function.

In our exercise, the rule is applied by first identifying each function’s role - outer and inner. The outer function is \( u^{3/2} \), and the inner is \( u = 3s^7 - 7s \). First, differentiate the outer function with respect to the inner function \( u \): This results in \( \frac{3}{2}u^{1/2} \). Next, differentiate the inner function with respect to \( s \), yielding \( 21s^6 - 7 \).

Applying the chain rule means combining these two results: \( g'(s) = \frac{3}{2}(3s^7 - 7s)^{1/2} \cdot (21s^6 - 7) \). This allows you to find the derivative of the overall composite function neatly.

Derivatives of Power Functions

Understanding the derivatives of power functions is key when tackling problems involving powers of variables. The power function rule states that if you have a function \( u^n \), its derivative is \( nu^{n-1} \).

In the problem given, the outer function is in the form \( u^{3/2} \). Differentiating it according to the power rule gives \( \frac{3}{2}u^{1/2} \), because you bring down the exponent as a coefficient and subtract one from the original exponent. It's straightforward but vital since it lays the foundation for using the chain rule effectively.

Being comfortable with the power rule and knowing how to apply it to various exponents makes finding derivatives, like the one in this exercise, much smoother. It helps in simplifying the process, making these computations less daunting and more intuitive.