Problem 45 - Calculus for Biology and Medicine

Question

Find the derivative with respect to the independent variable. $$ g(x)=\frac{1}{\sin \left(3 x^{2}-1\right)} $$

Step-by-Step Solution

Answer

The derivative is \( g'(x) = -\frac{6x \cdot \cos(3x^2 - 1)}{\sin^2(3x^2 - 1)} \).

Step 1: Identify the Derivative Formula

To find the derivative of a function of the form \( g(x) = \frac{1}{f(x)} \), we use the derivative formula for a reciprocal function: \( g'(x) = -\frac{f'(x)}{(f(x))^2} \). Here, \( f(x) = \sin(3x^2 - 1) \).

Step 2: Differentiate the Inner Function

Find the derivative of the inner function \( f(x) = \sin(3x^2 - 1) \). This requires using the chain rule. The derivative of \( \sin(u) \) is \( \cos(u) \cdot u' \). Here, \( u = 3x^2 - 1 \), so \( u' = 6x \). Thus, \( f'(x) = \cos(3x^2 - 1) \cdot 6x \).

Step 3: Substitute into the Derivative Formula

Substitute \( f'(x) \) and \( f(x) \) into the reciprocal derivative formula: \( g'(x) = -\frac{\cos(3x^2 - 1) \cdot 6x}{(\sin(3x^2 - 1))^2} \).

Step 4: Simplify the Expression

Write the expression in its simplest form. The derivative of the function is given by: \[ g'(x) = -\frac{6x \cdot \cos(3x^2 - 1)}{\sin^2(3x^2 - 1)} \].

Key Concepts

Understanding the Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. Let's break it down to understand how it works fully. Imagine a function composed of two or more other functions; the chain rule helps us find the derivative of such functions by working from the outside inwards. The chain rule states that if you have a composite function where one function, say \( f(g(x)) \), is inside another, you can differentiate it as follows:

• First, find the derivative of the outer function evaluated at the inner function, \( f'(g(x)) \).
• Then multiply this by the derivative of the inner function, \( g'(x) \).

So, the chain rule formula can be written as: \[( f(g(x)) )' = f'(g(x)) \cdot g'(x)\]In our exercise, the chain rule was needed to differentiate the inner function \( f(x) = \sin(3x^2 - 1) \). Here, the 'inner' function is \( u = 3x^2 - 1 \) with its derivative \( u' = 6x \). Then, applying the chain rule, we multiply \( \cos(u) \) by \( u' \) to get \( f'(x) = \cos(3x^2 - 1) \cdot 6x \). This process simplifies the differentiation of complex functions.

The Reciprocal Function Derivative

Reciprocal functions are quite interesting and require special attention when differentiating. A reciprocal function has the form \( g(x) = \frac{1}{f(x)} \), and its derivative must account for this special structure.To differentiate such functions, we use the formula for derivatives of reciprocal functions:\[g'(x) = -\frac{f'(x)}{(f(x))^2}\]This formula is crucial as it helps simplify the process by incorporating the negative sign and dividing by the square of the original function, \( (f(x))^2 \).In the exercise, we applied this formula to the reciprocal of the sine function, \( f(x) = \sin(3x^2 - 1) \). By first finding \( f'(x) \), which is \( \cos(3x^2 - 1) \cdot 6x \), and then using the reciprocal derivative formula, we achieved:\[g'(x) = -\frac{6x \cdot \cos(3x^2 - 1)}{\sin^2(3x^2 - 1)}\]This step-by-step approach ensures accuracy and clarity when handling derivatives of reciprocal functions.

The Art of Trigonometric Differentiation

Trigonometric differentiation comes into play when dealing with functions involving trigonometric expressions like sine, cosine, and tangent. Understanding these techniques is essential for solving complex calculus problems.The derivatives of basic trigonometric functions help set the foundation:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \(-\sin(x) \).
• For \( \tan(x) \), the derivative is \( \sec^2(x) \).

In our example, the use of \( \sin(3x^2 - 1) \) required us to apply the derivative of the sine function. We specifically needed to derive \( \sin(u) \) where \( u = 3x^2 - 1 \). Applying this, the derivative was found to be \( \cos(u) \), resulting in the differential step:\[f'(x) = \cos(3x^2 - 1) \cdot 6x\]Trigonometric differentiation allows us to tackle these kinds of complex expressions efficiently. By understanding these derivatives, you can handle various trigonometric combinations with ease.