Problem 4 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=-\sin x+\cos x-3 \csc x $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -\cos x + \sin x + 3 \csc x \cot x \).

Step 1: Identify the Function

The given function is \( f(x) = -\sin x + \cos x - 3 \csc x \). We need to find the derivative with respect to \( x \).

Step 2: Apply Derivative Rules

We will use the derivative rules: the derivative of \( \sin x \) is \( \cos x \), the derivative of \( \cos x \) is \( -\sin x \), and the derivative of \( \csc x \) is \( -\csc x \cot x \).

Step 3: Differentiate Each Term

Calculate the derivative of each term separately:- The derivative of \( -\sin x \) is \( -\cos x \).- The derivative of \( \cos x \) is \( -(-\sin x) = \sin x \).- The derivative of \( -3 \csc x \) is \( 3 \csc x \cot x \).

Step 4: Combine the Derivatives

Combine all the derivatives from each term:\[f'(x) = -\cos x + \sin x + 3 \csc x \cot x.\]

Step 5: Verify the Result

Review each derivative step to ensure correctness, checking that the signs and coefficients match. The final expression for the derivative is correct.

Key Concepts

Trigonometric Functions

Trigonometric functions like sine (\(\sin x\)), cosine (\(\cos x\)), and cosecant (\(\csc x\)) are foundational in mathematics. These functions relate to the angles and sides of triangles, primarily in right-angled triangles, but they also extend into periodic functions, which repeat in a regular pattern over intervals. The sine function represents the y-coordinate of a point on the unit circle, traversed counter-clockwise from the positive x-axis. On the other hand, the cosine function represents the x-coordinate. The cosecant function is the reciprocal of the sine function, defined as \(\csc x = \frac{1}{\sin x}\).

• The sine function has a range of \([-1, 1]\), and it is an odd function, meaning \(\sin(-x) = -\sin x\).
• The cosine function also ranges from \([-1, 1]\) and is an even function, so \(\cos(-x) = \cos x\).
• Cosecant has no range limitations like sine and cosine; however, it is undefined where sine is zero.

Understanding these properties of trigonometric functions is crucial for calculations in calculus and analyzing periodic behaviors in different contexts.

Differentiation Rules

Differentiation is the process of finding a derivative, which gives the rate at which a function is changing at any given point. For trigonometric functions, specific rules provide these rate changes quickly. For instance:

• The derivative of \(\sin x\) is \(\cos x\).
• The derivative of \(\cos x\) is \(-\sin x\).
• The derivative of \(\csc x\) is \(-\csc x \cot x\), considering \(\cot x\) is the cotangent function, another reciprocal trigonometric function defined as \(\cot x = \frac{\cos x}{\sin x}\).

These rules stem from fundamental calculus principles. To differentiate a function like \(f(x) = -\sin x + \cos x - 3\csc x\), you apply the differentiation rules to each component individually. This process becomes straightforward once you understand and remember these basic derivative properties of trigonometric functions.

Calculus

Calculus is a branch of mathematics that studies how things change. It deals with two primary concepts: differentiation and integration. Differentiation involves finding the derivative, while integration deals with finding the area under curves. Both of these methods are tools to solve problems involving change and motion.
In the context of finding the derivative of functions, calculus allows us to understand how a function behaves near any given point. Differentiation gives insights into several key aspects:

• The slope of the tangent line to the curve, indicating how steep the curve is at a particular point.
• The rate of change of the function values, which is useful in various physics and engineering problems.
• Critical points of a function, which help identify maximum, minimum, or inflection points.

When working with functions that include trigonometric components, using calculus simplifies finding rate of changes. It helps us create models for wave patterns, oscillations, and much more by combining differentiation rules with trigonometric identities. Through calculus, adjustments and precise solutions are possible for complex scenarios involving constant and variable changes.