Problem 61
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
Verified Answer
Using the derivative definition and the trigonometric identity, we find that \( \frac{d}{d x} \cos x = -\sin x \).
1Step 1: Define the Derivative of Cosine
Using the definition of the derivative, we have \[ \frac{d}{dx} \cos x = \lim_{h \to 0} \frac{\cos(x+h) - \cos(x)}{h}. \]
2Step 2: Apply the Trigonometric Identity
Substitute the identity for cosine of a sum into the expression: \[ \cos(x+h) = \cos x \cos h - \sin x \sin h. \] Substitute this into the derivative limit: \[ \frac{d}{dx} \cos x = \lim_{h \to 0} \frac{(\cos x \cos h - \sin x \sin h) - \cos x}{h}. \]
3Step 3: Simplify the Expression
Factoring out \(\cos x\), the expression becomes: \[ \frac{d}{dx} \cos x = \lim_{h \to 0} \frac{\cos x (\cos h - 1) - \sin x \sin h}{h}. \] Split the limit into two separate limits: \[ \frac{d}{dx} \cos x = \cos x \lim_{h \to 0} \frac{\cos h - 1}{h} - \sin x \lim_{h \to 0} \frac{\sin h}{h}. \]
4Step 4: Evaluate the Limits
Using standard limit results: \[ \lim_{h \to 0} \frac{\cos h - 1}{h} = 0 \] and \[ \lim_{h \to 0} \frac{\sin h}{h} = 1. \] Substitute these limits back into the expression: \[ \frac{d}{dx} \cos x = \cos x (0) - \sin x (1) = -\sin x. \]
5Step 5: Conclusion
Thus, using the definition of the derivative and the trigonometric identity, we have shown that \[ \frac{d}{d x} \cos x = -\sin x. \]
Key Concepts
Understanding DerivativesTrigonometric Identities in CalculusLimit Processes and Their Applications
Understanding Derivatives
Derivatives are a fundamental tool in calculus that help measure how a function changes as its input changes. In simple terms, the derivative provides the slope of the tangent line to the graph of a function at a given point. This concept is crucial in understanding rates of change and behavior of functions.
To find a derivative, we often use the limit definition of the derivative. Consider a function \( f(x) \). The derivative of \( f \) at a point \( x \) is defined as:
In the original exercise, we apply this definition to the cosine function, \( \cos(x) \), to derive its rate of change or derivative.
To find a derivative, we often use the limit definition of the derivative. Consider a function \( f(x) \). The derivative of \( f \) at a point \( x \) is defined as:
- \( \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)
In the original exercise, we apply this definition to the cosine function, \( \cos(x) \), to derive its rate of change or derivative.
Trigonometric Identities in Calculus
Trigonometric identities are essential equations that relate the angles and sides of a triangle. They simplify complex trigonometric expressions, making calculus problems easier to solve. The identity used in this exercise is:
In calculus, these identities help break down composite trigonometric expressions into simpler components. They are particularly useful when evaluating limits, as shown in the derivation of the derivative of \( \cos(x) \). By substituting \( \cos(x+h) \) using this identity, we can transform the original limit expression into a more manageable form for calculation.
- \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \).
In calculus, these identities help break down composite trigonometric expressions into simpler components. They are particularly useful when evaluating limits, as shown in the derivation of the derivative of \( \cos(x) \). By substituting \( \cos(x+h) \) using this identity, we can transform the original limit expression into a more manageable form for calculation.
Limit Processes and Their Applications
Limit processes are a cornerstone of calculus. They help describe the behavior of functions as the input of the function approaches a particular value, or in calculus terms, as \( h \to 0 \). This concept is prominently used in determining derivatives and integral calculations.
In the context of the derivative of \( \cos(x) \), we employ limits to handle the small increment \( h \) approaching zero. The two key limit results required here are:
Understanding limits is essential for grasping how differentiable functions change. They provide the gateway to finding derivatives, which in turn tell us about the instantaneous rates of changes of the functions.
In the context of the derivative of \( \cos(x) \), we employ limits to handle the small increment \( h \) approaching zero. The two key limit results required here are:
- \( \lim_{h \to 0} \frac{\cos h - 1}{h} = 0 \)
- \( \lim_{h \to 0} \frac{\sin h}{h} = 1 \)
Understanding limits is essential for grasping how differentiable functions change. They provide the gateway to finding derivatives, which in turn tell us about the instantaneous rates of changes of the functions.
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Problem 60
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