Derivative Calculator

Derivative of $\ln\left(\sin\left(x\right)\right)$

Differentiated with respect to x. Edit the function · Differentiation rules

Step-by-step solution

Input function
$$\ln\left(\sin\left(x\right)\right)$$
Start of the differentiation
$$\frac{d}{dx}{\left[\ln\left(\sin\left(x\right)\right)\right]}$$

Yellow: the expression being replaced. Green: its replacement.

Step 1 — Chain rule for the natural logarithm Differentiation rule
📖 Rule
$$\frac{d}{dx}\big(\ln({u})\big) = \frac{u'}{{u}}$$
With:
  • $u = \sin\left(x\right)$

Why: The outer derivative of the natural logarithm is the reciprocal of its argument. The chain rule adds the derivative of the argument in the numerator.

How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\ln\left(\sin\left(x\right)\right)\right]}\,}}}$$
✨ After: Chain rule for the natural logarithm
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{\frac{d}{dx}{\left[\sin\left(x\right)\right]}}{\sin\left(x\right)}\,}}}$$
Step 2 — Derivative of Sine Differentiation rule
📖 Rule
$$\frac{d}{dx}\sin\left(x\right)=\cos\left(x\right)$$

Why: The argument is exactly x. Apply the known elementary derivative.

Changed part: $\frac{d}{dx}{\left[\sin\left(x\right)\right]} \;\Longrightarrow\; \cos\left(x\right)$
How does this rule work? Wikipedia
🧮 Current expression
$$\frac{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}\,}}}}{\sin\left(x\right)}$$
✨ After: Derivative of Sine
$$\frac{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\cos\left(x\right)\,}}}}{\sin\left(x\right)}$$
Final result
$$\frac{\cos\left(x\right)}{\sin\left(x\right)}$$
Valid where:
  • $\sin\left(x\right)>0$ – The logarithm argument must be positive.

The derivative is only valid where the original function is defined.

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Cross-check with a computer algebra system

The derivative computed directly by the computer algebra system Maxima:

$$\frac{\cos\left(x\right)}{\sin\left(x\right)}$$
Step-by-step result ↔ Maxima: identical ✓