Questions: Which of the following expressions is the definition of the derivative of f(x)=cos(x) at the point (2, cos(2))? lim (h -> 0) (cos(x+h)-cos(x))/h lim (h -> 0) (cos(2+h)-cos(h))/h lim (h -> 0) (cos(2+h)-cos(2))/h lim (h -> 0) (cos(2+h)+cos(2))/h

Which of the following expressions is the definition of the derivative of f(x)=cos(x) at the point (2, cos(2))?
lim (h -> 0) (cos(x+h)-cos(x))/h
lim (h -> 0) (cos(2+h)-cos(h))/h
lim (h -> 0) (cos(2+h)-cos(2))/h
lim (h -> 0) (cos(2+h)+cos(2))/h

Transcript text: Which of the following expressions is the definition of the derivative of $f(x)=\cos (x)$ at the point $(2, \cos (2))$ ? $\lim _{h \rightarrow 0} \frac{\cos (x+h)-\cos (x)}{h}$ $\lim _{h \rightarrow 0} \frac{\cos (2+h)-\cos (h)}{h}$ $\lim _{h \rightarrow 0} \frac{\cos (2+h)-\cos (2)}{h}$ $\lim _{h \rightarrow 0} \frac{\cos (2+h)+\cos (2)}{h}$

Solution

Solution Steps

To find the definition of the derivative of $ f(x) = \cos(x) $ at the point $ (2, \cos(2)) $, we need to use the limit definition of the derivative. The correct expression should evaluate the change in $ \cos(x) $ at $ x = 2 $ as $ h $ approaches 0.

Solution Approach

Identify the function $ f(x) = \cos(x) $.
Use the limit definition of the derivative: $ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $.
Substitute $ x = 2 $ into the definition.

Step 1: Identify the Function and Point

We are given the function $ f(x) = \cos(x) $ and the point $ (2, \cos(2)) $. We need to find the derivative of $ f(x) $ at $ x = 2 $.

Step 2: Use the Limit Definition of the Derivative

The limit definition of the derivative is: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] Substituting $ x = 2 $ into the definition, we get: \[ f'(2) = \lim_{h \to 0} \frac{\cos(2+h) - \cos(2)}{h} \]

Step 3: Simplify the Expression

Using the trigonometric identity for the derivative of $ \cos(x) $, we know: \[ \frac{d}{dx} \cos(x) = -\sin(x) \] Thus, the derivative at $ x = 2 $ is: \[ f'(2) = -\sin(2) \]

Step 4: Evaluate the Derivative

The value of the derivative at $ x = 2 $ is: \[ f'(2) = -\sin(2) \]

Final Answer

The correct expression for the definition of the derivative of $ f(x) = \cos(x) $ at the point $ (2, \cos(2)) $ is: \[ \boxed{\lim_{h \to 0} \frac{\cos(2+h) - \cos(2)}{h}} \] Thus, the answer is the third option.

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