Derivative Calculator

Derivative of $2^{x}$

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

Step-by-step solution

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

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

Step 1 — Chain Rule (exponential with constant base) Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(a^{u}\right)=a^{u}\,\ln(a)\,u'$$
With:
  • $a = 2$
  • $u = x$

Why: The base is constant and the exponent depends on x. Keep the power, then multiply by the natural logarithm of the base and by the derivative of the exponent.

How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[2^{x}\right]}\,}}}$$
✨ After: Chain Rule (exponential with constant base)
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2^{x}\,\ln\left(2\right)\,\frac{d}{dx}{\left[x\right]}\,}}}$$
Step 2 — 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
$$2^{x}\,\ln\left(2\right)\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\right]}\,}}}$$
✨ After: Derivative of a Variable
$$2^{x}\,\ln\left(2\right) \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}$$
Step 3 — 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.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2^{x}\,\ln\left(2\right) \cdot 1\,}}}$$
✨ After: Remove a factor of one
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2^{x}\,\ln\left(2\right)\,}}}$$
Step 4 — 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.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2^{x}\,\ln\left(2\right)\,}}}$$
✨ After: Order the factors
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\ln\left(2\right) \cdot 2^{x}\,}}}$$
Final result
$$\ln\left(2\right) \cdot 2^{x}$$
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Cross-check with a computer algebra system

The derivative computed directly by the computer algebra system Maxima:

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