What is the derivative of sin(ln x) + cos(ln x) with respect to x at x = e?
Finding the Derivative of Trigonometric Functions involving Logarithm
The question asks us to find the derivative of the function \(f(x) = \sin(\ln x) + \cos(\ln x)\) with respect to \(x\) and then evaluate this derivative at a specific point, \(x = e\).
Steps to Solve the Derivative Problem
To find the derivative of \(f(x)\), we need to apply the rules of differentiation, specifically the chain rule, as the trigonometric functions involve a composite function (\(\ln x\)).
Step 1: Find the derivative of \(\sin(\ln x)\)
Using the chain rule, the derivative of \(\sin(u)\) with respect to \(x\) is \(\cos(u) \cdot \dfrac{du}{dx}\). Here, \(u = \ln x\). The derivative of \(\ln x\) with respect to \(x\) is \(\dfrac{1}{x}\).
So, the derivative of \(\sin(\ln x)\) is:
Step 2: Find the derivative of \(\cos(\ln x)\)
Using the chain rule, the derivative of \(\cos(u)\) with respect to \(x\) is \(-\sin(u) \cdot \dfrac{du}{dx}\). Here, \(u = \ln x\). The derivative of \(\ln x\) with respect to \(x\) is \(\dfrac{1}{x}\).
So, the derivative of \(\cos(\ln x)\) is:
Step 3: Find the derivative of the sum
The derivative of the sum of functions is the sum of their derivatives. So, the derivative of \(f(x) = \sin(\ln x) + \cos(\ln x)\) is:
We can factor out \(\dfrac{1}{x}\):
Step 4: Evaluate the derivative at \(x = e\)
Now, we need to substitute \(x = e\) into the derivative function \(f'(x)\). Recall that \(\ln e = 1\).
This is the value of the derivative of \(\sin(\ln x) + \cos(\ln x)\) with respect to \(x\) at \(x = e\).
Comparing with Options
Let's compare our calculated value with the given options:
- Option 1: \(\dfrac{\cos 1 - \sin 1}{e}\)
- Option 2: \(\dfrac{\sin 1 - \cos 1}{e}\)
- Option 3: \(\dfrac{\cos 1 + \sin 1}{e}\)
- Option 4: \(1\)
Our calculated value matches Option 1.
Revision Table: Key Concepts
| Concept | Description | Formula/Rule |
|---|---|---|
| Derivative of sin(u) | Derivative of sine function with argument u | \( \dfrac{d}{dx}(\sin u) = \cos u \cdot \dfrac{du}{dx} \) (Chain Rule) |
| Derivative of cos(u) | Derivative of cosine function with argument u | \( \dfrac{d}{dx}(\cos u) = -\sin u \cdot \dfrac{du}{dx} \) (Chain Rule) |
| Derivative of ln(x) | Derivative of the natural logarithm of x | \( \dfrac{d}{dx}(\ln x) = \dfrac{1}{x} \) |
| Chain Rule | Used for differentiating composite functions | \( \dfrac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x) \) |
| Evaluating Derivative | Finding the value of the derivative at a specific point \(x=a\) | Substitute \(a\) into the derivative function \(f'(x)\) to get \(f'(a)\) |
Additional Information: Natural Logarithm and Exponential Function
The natural logarithm, denoted as \(\ln x\) or \(\log_e x\), is the inverse function of the exponential function \(e^x\). The base of the natural logarithm is the mathematical constant \(e\), which is approximately 2.71828.
- Definition: \(y = \ln x\) if and only if \(x = e^y\).
- Key Property: \(\ln e = 1\) because \(e^1 = e\). This property was crucial in evaluating the derivative at \(x=e\).
- Domain of \(\ln x\): The natural logarithm is defined only for positive real numbers, i.e., \(x > 0\).
- Graph: The graph of \(y = \ln x\) increases slowly and passes through the point \((1, 0)\) since \(\ln 1 = 0\).
Understanding the relationship between the natural logarithm and the base \(e\) is fundamental for solving problems involving \(\ln x\) and \(e^x\) in calculus and other areas of mathematics.
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