Learn how to find the derivative of sin(ln x). The derivative is d/dx[sin(ln x)].
Below is the graph of f(x) = sin(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.
sin(ln x) is a composite function where ln x is the argument of the sine function. This requires using the Trigonometric Rule to differentiate the sine, combined with the chain rule to handle the natural logarithm inside.
💡 Why this works: The Trigonometric Rule applies to the sine function, and the chain rule accounts for the ln x. This composite structure dictates the method of differentiation for sin(ln x).
First, apply the Trigonometric Rule to sin(u), which gives cos(u), where u is ln x. Then, differentiate the inner function, ln x, using its derivative, which is 1/x.
💡 Why this works: Thus, the derivative of sin(ln x) is cos(ln x) times the derivative of ln x, yielding the final result of cos(ln x) * (1/x).
After applying the differentiation rules, we end up with cos(ln x) * (1/x). This is the simplified form of the derivative, which is the correct final result.
💡 Why this works: The simplified expression correctly represents the rate of change of sin(ln x) with respect to x, showing how the sine function changes when its argument is ln x.
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[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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