Problem 14 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=x^{2} \sec \frac{\pi}{6}+3 x \sec \frac{\pi}{4} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{4x}{\sqrt{3}} + 3\sqrt{2} \).

Step 1: Identify the Terms

The function given is \( f(x) = x^2 \sec \frac{\pi}{6} + 3x \sec \frac{\pi}{4} \). Here we need to differentiate two terms combined by addition.

Step 2: Determine the Constants

\( \sec \frac{\pi}{6} \) and \( \sec \frac{\pi}{4} \) are constants. Calculate their values: \( \sec \frac{\pi}{6} = 2/\sqrt{3} \) and \( \sec \frac{\pi}{4} = \sqrt{2} \). Substitute these into the function.

Step 3: Substitute the Constants Into the Function

The function with constants substituted is \( f(x) = x^2 \cdot \frac{2}{\sqrt{3}} + 3x \cdot \sqrt{2} \).

Step 4: Differentiate Each Term Separately

Differentiate \( x^2 \cdot \frac{2}{\sqrt{3}} \) with respect to \( x \) to get \( 2x \cdot \frac{2}{\sqrt{3}} = \frac{4x}{\sqrt{3}} \). Similarly, differentiate \( 3x \cdot \sqrt{2} \) to get \( 3 \cdot \sqrt{2} \).

Step 5: Combine the Derived Terms

Combine the results of differentiation to obtain the overall derivative: \( f'(x) = \frac{4x}{\sqrt{3}} + 3\sqrt{2} \).

Key Concepts

Constants in Calculus

In calculus, constants play an essential role, especially when differentiating functions. Constants are fixed values that do not change. In the exercise provided, constants appear as part of trigonometric functions like \( \sec \frac{\pi}{6} \) and \( \sec \frac{\pi}{4} \). Calculating these constants gives definite values \( \frac{2}{\sqrt{3}} \) and \( \sqrt{2} \) respectively. Understanding constants allows you to simplify the function before differentiation. Always remember, during differentiation, a constant multiplied by a function remains constant, allowing us to focus mainly on the variable part of the expression for differentiation.

Function Derivatives

Differentiation is a fundamental concept in calculus, focusing on how a function changes at any given point. It involves finding the derivative, which represents the slope of the function at a point or the function's rate of change. In the provided example, the task is to find the derivative of the function \( f(x) = x^2 \sec \frac{\pi}{6} + 3x \sec \frac{\pi}{4} \). Differentiating involves taking each term separately:

• For \( x^2 \), the derivative is \( 2x \).
• For a constant multiplied by \( x^2 \), like \( \frac{2}{\sqrt{3}} x^2 \), it becomes \( \frac{4x}{\sqrt{3}} \).
• For the term \( 3x \cdot \sqrt{2} \), the derivative is \( 3\sqrt{2} \), as the derivative of \( x \) is \( 1 \).

By individually differentiating each term and combining them, we form the complete derivative \( f'(x) = \frac{4x}{\sqrt{3}} + 3\sqrt{2} \). This derivative represents the rate at which \( f(x) \) changes with respect to \( x \).

Mathematical Notation

Mathematical notation is the language of mathematics, providing a concise way to express complex ideas. Understanding and correctly using notation is crucial in calculus, particularly with functions and derivatives. In the given example, the function \( f(x) \) denotes that \( f \) is a function with the independent variable \( x \). Differentiation is indicated by the symbol \( f'(x) \), which denotes the derivative of the function \( f \) with respect to \( x \).Functions like \( \sec \frac{\pi}{6} \) are part of trigonometric notations that define specific angles. In differentiation, it’s essential to recognize these symbols as constants or variables, as incorrect interpretation could lead to errors in derivative calculation. Using consistent notation like \( \frac{\underline{\phantom{xx}}}{\underline{\phantom{xx}}} \) for fractions helps in maintaining clarity, especially during algebraic manipulation.