Problem 12 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

\(f'(x) = 3x^2\)

Step 1: Express Constants

First, recognize that both \(\cos \frac{\pi}{3}\) and \(\cos \frac{\pi}{6}\) are constants. Calculate them: \(\cos \frac{\pi}{3} = \frac{1}{2}\) and \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\). The function becomes: \(f(x) = 2x^3 \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} = x^3 + \frac{\sqrt{3}}{2}\).

Step 2: Differentiate the Function

Differentiate the function \(f(x) = x^3 + \frac{\sqrt{3}}{2}\) with respect to \(x\). The derivative of \(x^3\) is \(3x^2\), and the derivative of a constant \(\frac{\sqrt{3}}{2}\) is 0. Therefore, \(f'(x) = 3x^2\).

Key Concepts

Understanding Calculus

Calculus is a branch of mathematics that focuses on change. It is primarily divided into two areas: differentiation and integration. Differentiation involves finding a derivative, which is a measure of how a function changes as its input changes. In this exercise, differentiation is used to find the rate of change of the function
\( f(x) = x^3 + \frac{\sqrt{3}}{2} \).
Calculus is essential for understanding and modeling real-world situations where change is involved. It is used in fields such as physics, engineering, economics, and statistics. Learning calculus can help you grasp the fundamental principles of motion and growth. By examining how small changes in the input of a function can lead to variations in the output, calculus provides invaluable insights for solving complex problems.

Exploring Derivative Rules

Derivative rules are critical for differentiating various types of functions. They provide formulas that streamline the process of finding derivatives. One basic rule is the power rule. This rule states that the derivative of \(x^n\) is \(nx^{n-1}\), where \(n\) is a constant.
In the step-by-step solution, the power rule is applied to differentiate \(x^3\) which gives \(3x^2\). Here's how it works:

• Write down the original exponent of the variable (3 in this case).
• Multiply the entire term by this exponent: \(3 \times x^3\).
• Subtract one from the original exponent to lower its power: \(x^{3-1} = x^2\).

Another important aspect of derivative rules is handling constant values. When differentiating constant terms, their derivative is always 0 because constants don’t change.
Whether you're differentiating a polynomial, a trigonometric function, or a product of functions, derivative rules help make the process more manageable.

Trigonometric Constants Explained

Trigonometric constants are the fixed values that come from evaluating trigonometric functions at specific angles. Common angles like \(\frac{\pi}{3}\) and \(\frac{\pi}{6}\) have well-known cosine values.
In our exercise, these constants are used as follow:

• \(\cos \frac{\pi}{3} = \frac{1}{2}\)
• \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\)

These constants help simplify problems, allowing us to express functions without fractional angles. In the solution, recognizing these constants simplifies the function, making differentiation straightforward.
By knowing these constants, you save time during calculations and improve accuracy. Trigonometric constants are a convenient tool in calculus when dealing with periodic functions and wave-related problems.