Problem 11 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \sqrt{3} x \).

Step 1: Understand the Function

We need to differentiate the function \( f(x) = x^{2} \sin \frac{\pi}{3} + \tan \frac{\pi}{4} \) with respect to \( x \). Note that \( \sin \frac{\pi}{3} \) and \( \tan \frac{\pi}{4} \) are constants since they are values of trigonometric functions evaluated at specific angles.

Step 2: Evaluate Constant Trigonometric Values

Calculate \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \) and \( \tan \frac{\pi}{4} = 1 \). Substitute these values into the function: \( f(x) = x^{2} \times \frac{\sqrt{3}}{2} + 1 \).

Step 3: Rewrite the Function

Rewrite the function using the evaluated constants: \( f(x) = \frac{\sqrt{3}}{2} x^{2} + 1 \).

Step 4: Differentiate the Function

Differentiate \( f(x) = \frac{\sqrt{3}}{2} x^{2} + 1 \) with respect to \( x \). The derivative of \( \frac{\sqrt{3}}{2} x^{2} \) is \( \frac{\sqrt{3}}{2} \times 2x \) and the derivative of \( 1 \) is \( 0 \).

Step 5: Simplify the Derivative

Simplify the derivative expression: \( \frac{\sqrt{3}}{2} \times 2x = \sqrt{3} x \). Thus, \( f'(x) = \sqrt{3} x \).

Key Concepts

Understanding Trigonometric Functions in Calculus

In calculus, trigonometric functions like sine, cosine, and tangent are just as crucial as algebraic functions for analyzing and understanding changes. These functions describe relationships between angles and sides of triangles and are used extensively in solving calculus problems.
In the given exercise, we need to differentiate a function that includes trigonometric constants. Specifically, we encounter constants such as \( \sin \frac{\pi}{3} \) and \( \tan \frac{\pi}{4} \). Understanding these functions inside out helps in appreciating how these constants are used in calculus.
Why are these constants? Because these trigonometric functions are evaluated at specific angles.

• \( \sin \frac{\pi}{3} \) refers to the sine of 60 degrees, which is always \( \frac{\sqrt{3}}{2} \).
• \( \tan \frac{\pi}{4} \) is the tangent of 45 degrees, always equal to 1.

Knowing these values by heart can help speed up solving calculus problems, saving the extra step of calculation.

Evaluating Constants in Calculus Problems

In calculus, it’s common to evaluate constants as a part of simplifying expressions before taking derivatives. When dealing with differentiation, constants have a straightforward behavior. They remain unchanged when the derivative is calculated, which simplifies the calculus process.
For example, if you're differentiating \( f(x) = x^2 \sin \frac{\pi}{3} + \tan \frac{\pi}{4} \), it is crucial to first evaluate these as constants:

• Calculate \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \).
• Calculate \( \tan \frac{\pi}{4} = 1 \).

By replacing \( \sin \frac{\pi}{3} \) with \( \frac{\sqrt{3}}{2} \) and \( \tan \frac{\pi}{4} \) with 1, the function becomes easier to differentiate:\[ \frac{\sqrt{3}}{2}x^2 + 1 \]By understanding and applying constant evaluations properly, the differentiation process becomes much more streamlined and less error-prone.

Basic Steps in Calculus Problem Solving

Solving calculus problems often involves a series of basic steps that, when followed correctly, make differentiation straightforward. Let's break down the approach used in differentiating the function \( f(x) = x^2 \sin \frac{\pi}{3} + \tan \frac{\pi}{4} \):

• First, simplify the function by evaluating constant values as mentioned earlier. This simplifies the expression \( f(x) = \frac{\sqrt{3}}{2}x^2 + 1 \).
• Next, apply basic rules of differentiation. For instance, the power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). Here, the derivative of \( \frac{\sqrt{3}}{2}x^2 \) is \( \frac{\sqrt{3}}{2} \times 2x \).
• Lastly, differentiate the constant term. The derivative of a constant is always 0, so it disappears from the derivative expression.

The derivative simplifies to \( \sqrt{3}x \). By logically following these steps—evaluating constants, applying differentiation rules, and simplifying—you can solve calculus problems like a pro!