Problem 61 - Calculus for Biology and Medicine

Question

Use the identity $$ \cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta $$ and the definition of the derivative to show that $$ \frac{d}{d x} \cos x=-\sin x $$

Step-by-Step Solution

Answer

The derivative of \( \cos x \) is \( -\sin x \) using the trigonometric identity and the derivative definition.

Step 1: Recall the Definition of Derivative

The derivative of a function \( f(x) \) is defined by the limit \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). We will use this definition to find the derivative of \( \cos x \).

Step 2: Apply the Definition to \( \cos x \)

Substitute \( f(x) = \cos x \) into the derivative definition: \[ \frac{d}{dx} \cos x = \lim_{h \to 0} \frac{\cos(x+h) - \cos x}{h} \]

Step 3: Use the Angle Addition Identity

Apply the identity \( \cos(x+h) = \cos x \cos h - \sin x \sin h \) to the expression in the numerator: \[ \lim_{h \to 0} \frac{(\cos x \cos h - \sin x \sin h) - \cos x}{h} \]

Step 4: Simplify the Expression

Simplify the numerator: \( \cos x \cos h - \cos x = \cos x (\cos h - 1) \), so: \[ \lim_{h \to 0} \frac{\cos x (\cos h - 1) - \sin x \sin h}{h} \]

Step 5: Split the Limit

Separate into two separate limits: \[ \lim_{h \to 0} \frac{\cos x (\cos h - 1)}{h} - \lim_{h \to 0} \frac{\sin x \sin h}{h} \]

Step 6: Evaluate the Limits

As \( h \to 0 \), the limit \( \lim_{h \to 0} \frac{\cos h - 1}{h} = 0 \) and \( \lim_{h \to 0} \frac{\sin h}{h} = 1 \). Thus, \( \cos x \cdot 0 = 0 \) and \( -\sin x \cdot 1 = -\sin x \).

Step 7: Conclude the Derivative Calculation

Combine the results to show that \( \frac{d}{dx} \cos x = -\sin x \).

Key Concepts

Derivative of cosine

When studying calculus, understanding how to find derivatives of trigonometric functions is crucial. One of the most fundamental derivatives is that of the cosine function, written as \( \frac{d}{dx} \cos x \). At first glance, it might be surprising to find that the derivative of cosine is actually related to the sine function. Specifically, the derivative of \( \cos x \) is \( -\sin x \).This negative sign is vital and indicates how the rate of change of the cosine function behaves as \( x \) changes. Cosine represents the horizontal component of a unit circle on the Cartesian plane. When differentiating, we observe how this horizontal projection changes, which results in the negative sine function. This reflects the phase shift between sine and cosine in trigonometric cycles. Understanding this derivative is pivotal for anyone transitioning from basic calculus to more advanced topics, as it sets the foundation for tackling complex problems involving trigonometric functions.

Limit definition of derivative

In calculus, the derivative of a function at a given point is rigorously defined using limits. The limit definition of a derivative is given as:\[\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]This represents the slope of the tangent line to the curve of the function at that specific point. It's the backbone of derivatives and helps in precisely calculating the instantaneous rate of change.For the cosine function, this means using the expression:\[\frac{d}{dx} \cos x = \lim_{h \to 0} \frac{\cos(x+h) - \cos x}{h}\]Substituting and evaluating this limit using trigonometric identities leads us to the correct derivative. The interplay between limits and trigonometric functions is fascinating, as it requires a deep understanding of both disciplines to proceed with computations. This concept ensures that the derivative calculated is exact and enforces a mechanism to handle even the tiniest changes in \( x \).

Trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for all angles. They simplify the following of derivatives and integrals, especially for functions such as sine, cosine, and tangent. These identities can be used creatively to transform expressions into more manageable forms.In the case of differentiating \( \cos x \), the angle addition identity is crucial. According to this identity:\[\cos(\alpha+\beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\]This allows us to rewrite the expression \( \cos(x+h) \) in terms of \( x \) and \( h \). By substituting it into the limit definition of a derivative, it simplifies the process of finding the derivative of cosine. Without these identities, simplifying the derivative expressions would become significantly more cumbersome. Therefore, mastering these identities is key to tackling various problems in calculus and beyond.