Derivative Calculator

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

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

Step-by-step solution

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

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

Step 1 — General Power Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(u^{v}\right)=u^{v}\,\big(v'\,\ln(u)+\frac{v\,u'}{u}\big)$$
With:
  • $u = x$
  • $v = \sin\left(x\right)$

Why: Both the base and the exponent depend on x. Both changes contribute to the derivative. This real formula assumes a positive base.

How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{\sin\left(x\right)}\right]}\,}}}$$
✨ After: General Power Rule
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x^{\sin\left(x\right)}\,\left(\frac{d}{dx}{\left[\sin\left(x\right)\right]}\,\ln\left(x\right) + \sin\left(x\right) \cdot \frac{\frac{d}{dx}{\left[x\right]}}{x}\right)\,}}}$$
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
$$x^{\sin\left(x\right)}\,\left(\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}\,}}}\,\ln\left(x\right) + \sin\left(x\right) \cdot \frac{\frac{d}{dx}{\left[x\right]}}{x}\right)$$
✨ After: Derivative of Sine
$$x^{\sin\left(x\right)}\,\left(\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\cos\left(x\right)\,}}}\,\ln\left(x\right) + \sin\left(x\right) \cdot \frac{\frac{d}{dx}{\left[x\right]}}{x}\right)$$
Step 3 — 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
$$x^{\sin\left(x\right)}\,\left(\cos\left(x\right)\,\ln\left(x\right) + \sin\left(x\right) \cdot \frac{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\right]}\,}}}}{x}\right)$$
✨ After: Derivative of a Variable
$$x^{\sin\left(x\right)}\,\left(\cos\left(x\right)\,\ln\left(x\right) + \sin\left(x\right) \cdot \frac{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}{x}\right)$$
Step 4 — Combine into one fraction Simplification
📖 Rule
$$a\cdot\frac{b}{c}=\frac{a\cdot b}{c}$$

Why: A factor in front of a fraction multiplies its numerator, giving a single fraction.

Changed part: $\sin\left(x\right) \cdot \frac{1}{x} \;\Longrightarrow\; \frac{\sin\left(x\right)}{x}$
🧮 Current expression
$$x^{\sin\left(x\right)}\,\left(\cos\left(x\right)\,\ln\left(x\right) + \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\sin\left(x\right) \cdot \frac{1}{x}\,}}}\right)$$
✨ After: Combine into one fraction
$$x^{\sin\left(x\right)}\,\left(\cos\left(x\right)\,\ln\left(x\right) + \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{\sin\left(x\right)}{x}\,}}}\right)$$
Final result
$$x^{\sin\left(x\right)}\,\left(\cos\left(x\right)\,\ln\left(x\right) + \frac{\sin\left(x\right)}{x}\right)$$
Valid where:
  • $x>0$ – The general real power rule is applied on a positive base.

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:

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