Learn how to find the derivative of cos(ln x). The derivative is d/dx[cos(ln x)].
Below is the graph of f(x) = cos(ln x) and its derivative f'(x). Notice how the original function and derivative relate to each other.
Original Function:
Derivative:
This derivative represents the rate of change of the original function. The calculation follows standard differentiation rules as shown in the step-by-step solution below.
To differentiate cos(ln x), we first identify it as a composition of two functions: cosine and the natural logarithm of x. The outer function is cosine, and the inner function is ln x.
💡 Why this works: Since cos(ln x) involves a composition, we apply the Chain Rule. Additionally, because cosine is a trigonometric function, we will use the Trigonometric Rule for differentiation.
First, differentiate the outer function, cos(u), where u = ln x. The derivative of cos(u) is -sin(u). Then, differentiate the inner function, ln x, which has the derivative 1/x.
💡 Why this works: Applying the Chain Rule, the derivative of cos(ln x) becomes -sin(ln x) multiplied by the derivative of ln x, which is 1/x. So, we have: d/dx[cos(ln x)] = -sin(ln x) * (1/x).
The derivative we obtained is -sin(ln x) * (1/x), which expresses how the function cos(ln x) changes with respect to x. To verify this, we check that both the trigonometric and logarithmic parts were correctly differentiated.
💡 Why this works: This derivative is correct because it accurately reflects the application of the Chain Rule and the differentiation of both cosine and the natural logarithm.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ Common Mistake:
Confusing the derivatives of sin(x) and cos(x)
✅ Correct Approach:
Remember: d/dx[sin(x)] = cos(x), d/dx[cos(x)] = -sin(x)
❌ Common Mistake:
Not simplifying the final answer
✅ Correct Approach:
Always simplify your derivative to its most reduced form
Double-check your work by comparing your steps with our solution. Use our calculator to verify each step of the differentiation process.
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