Derivative Calculator

Derivative of $x^{2}\,\sin\left(x\right)$

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

Step-by-step solution

Input function
$$x^{2}\,\sin\left(x\right)$$
Start of the differentiation
$$\frac{d}{dx}{\left[x^{2}\,\sin\left(x\right)\right]}$$

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

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

Why: First, differentiate the first factor and multiply by the unchanged second factor. Add the unchanged first factor times the derivative of the second. Do not simply multiply the two derivatives.

How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2}\,\sin\left(x\right)\right]}\,}}}$$
✨ After: Product Rule
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2}\right]}\,\sin\left(x\right) + x^{2}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}\,}}}$$
Step 2 — Power Rule (special case) Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}$$
With:
  • $u = x$
  • $c = 2$

Why: Bring the current exponent down as a multiplier and subtract one from the exponent. Evaluate that small calculation in a separate step.

Changed part: $\frac{d}{dx}{\left[x^{2}\right]} \;\Longrightarrow\; 2\,x^{2 - 1}$
How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2}\right]}\,}}}\,\sin\left(x\right) + x^{2}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}$$
✨ After: Power Rule (special case)
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x^{2 - 1}\,}}}\,\sin\left(x\right) + x^{2}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}$$
Step 3 — Evaluate numbers Simplification
📖 Rule

Why: Evaluate only the highlighted numeric expression, preserving exact fractions and decimals.

Changed part: $2 - 1 \;\Longrightarrow\; 1$
🧮 Current expression
$$2\,x^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 - 1\,}}}}\,\sin\left(x\right) + x^{2}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}$$
✨ After: Evaluate numbers
$$2\,x^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}\,\sin\left(x\right) + x^{2}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}$$
Step 4 — Power with exponent one Simplification
📖 Rule
$$a^1=a$$

Why: An expression to the first power is the expression itself.

Changed part: $x^{1} \;\Longrightarrow\; x$
🧮 Current expression
$$2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x^{1}\,}}}\,\sin\left(x\right) + x^{2}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}$$
✨ After: Power with exponent one
$$2\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x\,}}}\,\sin\left(x\right) + x^{2}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}$$
Step 5 — 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
$$2\,x\,\sin\left(x\right) + x^{2}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}\,}}}$$
✨ After: Derivative of Sine
$$2\,x\,\sin\left(x\right) + x^{2}\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\cos\left(x\right)\,}}}$$
Final result
$$2\,x\,\sin\left(x\right) + x^{2}\,\cos\left(x\right)$$
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Cross-check with a computer algebra system

The derivative computed directly by the computer algebra system Maxima:

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