Problem 6 - Calculus for Biology and Medicine

Question

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\sqrt{2 x+7} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{1}{\sqrt{2x + 7}} \).

Step 1: Rewrite the Function

The given function is \( f(x) = \sqrt{2x + 7} \). It is easier to differentiate if we rewrite it using a fractional exponent: \( f(x) = (2x + 7)^{1/2} \).

Step 2: Apply the Chain Rule

To differentiate \( f(x) = (2x + 7)^{1/2} \), use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \).

Step 3: Differentiate the Outer Function

The outer function for \( f(x) = (2x + 7)^{1/2} \) is \( u^{1/2} \) where \( u = 2x + 7 \). The derivative of \( u^{1/2} \) with respect to \( u \) is \( \frac{1}{2}u^{-1/2} \).

Step 4: Differentiate the Inner Function

The inner function is \( u = 2x + 7 \). The derivative with respect to \( x \) is \( 2 \).

Step 5: Apply the Chain Rule to Find the Derivative

Using the chain rule, multiply the derivative of the outer function by the derivative of the inner function: \( f'(x) = \frac{1}{2}(2x+7)^{-1/2} \times 2 = (2x+7)^{-1/2} \).

Step 6: Rewrite the Derivative in Terms of a Radical

The derivative \( f'(x) = (2x+7)^{-1/2} \) can be rewritten using a radical: \( f'(x) = \frac{1}{\sqrt{2x + 7}} \).

Key Concepts

Chain Rule

The Chain Rule is an essential tool in calculus for differentiating composite functions. It comes into play when you have a function inside another function. For instance, in the expression \((2x + 7)^{1/2}\), there is a clear inner function, \(2x + 7\), and an outer function, \(u^{1/2}\).

• The Chain Rule helps us manage these layers by stating: the derivative of a composite function \(f(g(x))\) can be found by multiplying the derivative of the outer function \(f'(u)\) with the derivative of the inner function \(g'(x)\).
• In practical terms, you first differentiate the outer function with respect to the inner function, then differentiate the inner function with respect to the original variable.

Applying the chain rule ensures we account for changes in the inner function as they affect the entire composite structure. Understanding this principle is crucial when dealing with nested functions.

Fractional Exponents

Fractional exponents offer a convenient way to express roots and enhance the differentiation process. Let's take the example of \(\sqrt{2x + 7}\).

• Using fractional exponents, this expression becomes \((2x + 7)^{1/2}\). This conversion makes it easier to apply calculus operations such as differentiation.
• Fractional exponents follow the same rules as integer exponents: \(a^{m/n}\) denotes the \(n\)-th root of \(a\) raised to the \(m\)-th power.
• They also improve the process of finding derivatives using mechanisms like the Power Rule, which applies to any power of a variable.

Being able to switch between root notation and exponent notation is a valuable skill that provides flexibility in solving calculus problems.

Composite Functions

Composite functions are formed by inputting one function into another. This construction is symbolically written as \(f(g(x))\). For example, given \(f(x) = \sqrt{2x + 7}\), the composite nature is already evident.

• Understanding composite functions involves recognizing the role of both the inner \(g(x)\) and outer \(f(x)\) functions.
• These functions require careful application of differentiation techniques like the Chain Rule because changes in either part influence the whole.
• The outer function handles the overarching structure, while the inner function supplies the specific variable input details.

Mastering the art of breaking down composite functions into manageable parts and applying the appropriate rules allows you to tackle complex calculus problems efficiently.