Problem 11 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems 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 of the function is \( f'(x) = \sqrt{3}x \).

Step 1: Identify Constants and Functions

First, let's identify the constant and the functions within the given function \( f(x) = x^2 \sin \left( \frac{\pi}{3} \right) + \tan\left( \frac{\pi}{4} \right) \). Here, \( \sin \left( \frac{\pi}{3} \right) \) and \( \tan\left( \frac{\pi}{4} \right) \) are constants, while \( x^2 \) is a variable term.

Step 2: Calculate the Constant Values

Calculate the values of the trigonometric constants. We know from trigonometric tables that \( \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \) and \( \tan\left( \frac{\pi}{4} \right) = 1 \). Therefore, the function simplifies to \( f(x) = x^2 \cdot \frac{\sqrt{3}}{2} + 1 \).

Step 3: Differentiate Using Constant Multiple Rule and Power Rule

Differentiate the function \( f(x) = \frac{\sqrt{3}}{2} x^2 + 1 \) with respect to \( x \). Use the constant multiple rule, which says the derivative of \( c \cdot x^n \) is \( c \cdot n \cdot x^{n-1} \), and the power rule, which says the derivative of \( x^n \) is \( nx^{n-1} \). The derivative of \( 1 \), a constant, is \( 0 \). Applying these rules, we get: \( f'(x) = \frac{\sqrt{3}}{2} \cdot 2x + 0 \).

Step 4: Simplify the Derivative

Simplify the expression \( f'(x) = \frac{\sqrt{3}}{2} \cdot 2x \). The \( 2 \)s cancel out, leaving: \( f'(x) = \sqrt{3} x \). This is the derivative of the original function with respect to \( x \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in calculus, especially in problems involving differentiation and integration. In the original exercise, two trigonometric functions were involved: \( \sin \frac{\pi}{3} \) and \( \tan \frac{\pi}{4} \).

• \( \sin \) and \( \tan \) are among the basic trigonometric functions, which are used to relate the angles of a triangle to the lengths of its sides.
• The sine function, noted as \( \sin(\theta) \), gives the ratio of the opposite side to the hypotenuse in a right triangle.
• The tangent function, \( \tan(\theta) \), represents the ratio of the opposite side to the adjacent side.
• Specific angles, like \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \), have well-known values. For example, \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \) and \( \tan \frac{\pi}{4} = 1 \).

Understanding these values and properties is essential because once you know the trigonometric value of these angles, you can easily substitute and simplify expressions as we did in this exercise. The constants from these functions were key to transforming the expression into a simpler polynomial form that we could differentiate easily.

Constant Multiple Rule

The Constant Multiple Rule is a significant differentiation rule that simplifies finding the derivative of a function when it is multiplied by a constant. Here's how it works: if you have a function of the form \( c \cdot f(x) \), where \( c \) is a constant, the derivative is determined by \[\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)\].In our exercise:

• We needed to differentiate \( \frac{\sqrt{3}}{2} \cdot x^2 \). Here, \( \frac{\sqrt{3}}{2} \) serves as the constant, while \( x^2 \) is the function.
• The Constant Multiple Rule allows us to "move" the constant value to the front, focusing your differentiation efforts on the variable part.
• This rule applies widely and can often be combined with other rules, like the power rule, to efficiently take derivatives.

Using the Constant Multiple Rule made it straightforward to differentiate \( \frac{\sqrt{3}}{2} x^2 \), as seen in the step-by-step solution.

Power Rule

The Power Rule is one of the most simple yet powerful tools in calculus, used for finding the derivative of functions of the form \( x^n \). According to the Power Rule, \[\frac{d}{dx}[x^n] = nx^{n-1}\].Let's look at how it was used in the original problem:

• We applied the Power Rule to \( x^2 \), where \( n = 2 \).
• The rule tells us to multiply the term by the power: \( 2 \cdot x^{2-1} \), simplifying to \( 2x \).
• Combining this with the Constant Multiple Rule provides an efficient way to differentiate such polynomial expressions.
• After differentiating, the expression \( \frac{\sqrt{3}}{2} \cdot 2x \) emerged, leading to the simplified derivative \( \sqrt{3}x \).

Thanks to the Power Rule, differentiating polynomial functions becomes a relatively straightforward task. It's a foundational concept that builds toward understanding more complex calculus operations.