Derivative Calculator

Derivative of $\frac{\sin\left(x\right)}{x}$

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

Step-by-step solution

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

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

Step 1 — Quotient Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^{2}}$$
With:
  • $u = \sin\left(x\right)$
  • $v = x$

Why: In the numerator, subtract the numerator times the derivative of the denominator from the derivative of the numerator times the denominator. Divide by the squared denominator. The original denominator must be nonzero.

How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\frac{\sin\left(x\right)}{x}\right]}\,}}}$$
✨ After: Quotient Rule
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{\frac{d}{dx}{\left[\sin\left(x\right)\right]}\,x - \sin\left(x\right)\,\frac{d}{dx}{\left[x\right]}}{x^{2}}\,}}}$$
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
$$\frac{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}\,}}}\,x - \sin\left(x\right)\,\frac{d}{dx}{\left[x\right]}}{x^{2}}$$
✨ After: Derivative of Sine
$$\frac{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\cos\left(x\right)\,}}}\,x - \sin\left(x\right)\,\frac{d}{dx}{\left[x\right]}}{x^{2}}$$
Step 3 — Order the factors Simplification
📖 Rule
$$u\cdot c=c\cdot u$$

Why: The order of factors may be changed (commutative law). Usual order: number, constants, powers of the variable, other factors.

Changed part: $\cos\left(x\right)\,x \;\Longrightarrow\; x\,\cos\left(x\right)$
🧮 Current expression
$$\frac{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\cos\left(x\right)\,x\,}}} - \sin\left(x\right)\,\frac{d}{dx}{\left[x\right]}}{x^{2}}$$
✨ After: Order the factors
$$\frac{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x\,\cos\left(x\right)\,}}} - \sin\left(x\right)\,\frac{d}{dx}{\left[x\right]}}{x^{2}}$$
Step 4 — Derivative of a Variable Differentiation rule
📖 Rule
$$\frac{d}{dx}(x)=1$$

Why: When x increases by one, the value of this term also increases by one. Its derivative is therefore one.

Changed part: $\frac{d}{dx}{\left[x\right]} \;\Longrightarrow\; 1$
How does this rule work?
🧮 Current expression
$$\frac{x\,\cos\left(x\right) - \sin\left(x\right)\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\right]}\,}}}}{x^{2}}$$
✨ After: Derivative of a Variable
$$\frac{x\,\cos\left(x\right) - \sin\left(x\right) \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}{x^{2}}$$
Step 5 — Remove a factor of one Simplification
📖 Rule
$$1\cdot a=a,\quad \frac{a}{1}=a$$

Why: Multiplication or division by one leaves the value unchanged.

Changed part: $\sin\left(x\right) \cdot 1 \;\Longrightarrow\; \sin\left(x\right)$
🧮 Current expression
$$\frac{x\,\cos\left(x\right) - \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\sin\left(x\right) \cdot 1\,}}}}{x^{2}}$$
✨ After: Remove a factor of one
$$\frac{x\,\cos\left(x\right) - \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\sin\left(x\right)\,}}}}{x^{2}}$$
Final result
$$\frac{x\,\cos\left(x\right) - \sin\left(x\right)}{x^{2}}$$
Valid where:
  • $x\ne0$ – The original denominator must be nonzero.

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:

$$\frac{\cos\left(x\right)}{x} - \frac{\sin\left(x\right)}{x^{2}}$$
Step-by-step result ↔ Maxima: identical ✓