Problem 13 - Calculus for Biology and Medicine

Question

Differentiate the functions given with respect to the independent variable. $$ f(x)=-3 x^{4} \tan \frac{\pi}{6}-\cot \frac{\pi}{6} $$

Step-by-Step Solution

Answer

The derivative is \(-4\sqrt{3} x^3\).

Step 1: Identify Constants and Variable Parts

In the given function \( f(x) = -3x^{4} \tan \frac{\pi}{6} - \cot \frac{\pi}{6} \), notice that \( \tan \frac{\pi}{6} \) and \( \cot \frac{\pi}{6} \) are constants because they are evaluated at a specific angle, \( \frac{\pi}{6} \). The only variable part is \( x^4 \).

Step 2: Evaluate Constants

Calculate the values of the trigonometric functions at \( \frac{\pi}{6} \):- \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \)- \( \cot \frac{\pi}{6} = \sqrt{3} \)

Step 3: Substitute Constant Values

Substitute the calculated constant values into the function:\[f(x) = -3x^{4} \cdot \frac{1}{\sqrt{3}} - \sqrt{3}\]Simplify to get:\[f(x) = -\frac{3}{\sqrt{3}} x^4 - \sqrt{3}\]

Step 4: Simplify the Function

Simplify \( -\frac{3}{\sqrt{3}} x^4 \):\[-\frac{3}{\sqrt{3}} x^4 = -\sqrt{3} x^4\]So, the function becomes:\[f(x) = -\sqrt{3} x^4 - \sqrt{3}\]

Step 5: Differentiate the Function

Differentiate each term of the simplified function with respect to \( x \):- The derivative of \(-\sqrt{3} x^4 \) is: \[-\sqrt{3} \cdot 4 x^{3} = -4\sqrt{3} x^3\]- The derivative of the constant \(-\sqrt{3} \) is 0 as the derivative of a constant is always 0.So, the derivative of \( f(x) \) is:\[f'(x) = -4\sqrt{3} x^3\]

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles to side ratios in right-angled triangles. They are essential in calculus, especially when dealing with periodic phenomena. In the exercise, we encounter two trigonometric functions:

• \( \tan \frac{\pi}{6} \)
• \( \cot \frac{\pi}{6} \)

These functions evaluate specific angles. The tangent of an angle is the ratio of the opposite side to the adjacent side in a right triangle. Whereas, the cotangent is the reciprocal of the tangent, the ratio of the adjacent side to the opposite side. Here, \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \) and \( \cot \frac{\pi}{6} = \sqrt{3} \). The evaluation of these functions at the angle \( \frac{\pi}{6} \) yields constants, making it easier to differentiate the expression by focusing solely on the variable term \( x^4 \).

Constant Evaluation

Understanding the role of constants is vital in differentiation. Once trigonometric functions are evaluated at specific angles, they simplify to constants, which remain invariant even as other variables change. In our function \( f(x) = -3x^{4} \tan \frac{\pi}{6} - \cot \frac{\pi}{6} \), the expression contains constants derived from evaluating the trigonometric functions:

• \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \)
• \( \cot \frac{\pi}{6} = \sqrt{3} \)

Substituting these values allows us to simplify the expression to \( f(x) = -\frac{3}{\sqrt{3}} x^4 - \sqrt{3} \). This simplification is crucial for the differentiation process, as constants other than the variable vanish when taking derivatives. Differentiating constants results in zero, which streamlines the latter steps of calculus operations.

Power Rule for Differentiation

The power rule for differentiation is a basic yet powerful tool for finding the derivative of functions. This rule states that if you have a function of the form \( ax^n \), its derivative is \( anx^{n-1} \). In our exercise, the power rule is applied to \( -\sqrt{3} x^4 \).

• The coefficient, \( -\sqrt{3} \), remains unaffected as a constant factor.
• The power of \( x \) (which is 4) gets multiplied by this coefficient, and the new power is reduced by 1.

Thus, the derivative becomes \( -4\sqrt{3} x^3 \). Knowing this rule simplifies differentiation immensely. Constant terms like \( -\sqrt{3} \) differentiate to zero. Applying the power rule efficiently reduces the complexity of expressions, culminating in finding the derivative \( f'(x) = -4\sqrt{3} x^3 \).