Derivative Calculator

Derivative of $\tan\left(x\right)^{3}$

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

Step-by-step solution

Input function
$$\tan\left(x\right)^{3}$$
Start of the differentiation
$$\frac{d}{dx}{\left[\tan\left(x\right)^{3}\right]}$$

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

Step 1 — Power Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}\,u'$$
With:
  • $u = \tan\left(x\right)$
  • $c = 3$

Why: Differentiate the outer power: bring down the exponent and subtract one from it. The chain rule adds the derivative of the entire inner expression as another factor.

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

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

Changed part: $3 - 1 \;\Longrightarrow\; 2$
🧮 Current expression
$$3\,\tan\left(x\right)^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,3 - 1\,}}}}\,\frac{d}{dx}{\left[\tan\left(x\right)\right]}$$
✨ After: Evaluate numbers
$$3\,\tan\left(x\right)^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,}}}}\,\frac{d}{dx}{\left[\tan\left(x\right)\right]}$$
Step 3 — Derivative of Tangent Differentiation rule
📖 Rule
$$\frac{d}{dx}\tan\left(x\right)=\frac{1}{\cos\left(x\right)^{2}}$$

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

Changed part: $\frac{d}{dx}{\left[\tan\left(x\right)\right]} \;\Longrightarrow\; \frac{1}{\cos\left(x\right)^{2}}$
How does this rule work? Wikipedia
🧮 Current expression
$$3\,\tan\left(x\right)^{2}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\tan\left(x\right)\right]}\,}}}$$
✨ After: Derivative of Tangent
$$3\,\tan\left(x\right)^{2} \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\cos\left(x\right)^{2}}\,}}}$$
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.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,3\,\tan\left(x\right)^{2} \cdot \frac{1}{\cos\left(x\right)^{2}}\,}}}$$
✨ After: Combine into one fraction
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{3\,\tan\left(x\right)^{2}}{\cos\left(x\right)^{2}}\,}}}$$
Final result
$$\frac{3\,\tan\left(x\right)^{2}}{\cos\left(x\right)^{2}}$$
Valid where:
  • $\cos\left(x\right)\ne0$ – The function and its derivative are undefined at poles.

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:

$$3\,\sec\left(x\right)^{2}\,\tan\left(x\right)^{2}$$
Step-by-step result ↔ Maxima: identical ✓