Derivative Calculator

Derivative of $\sec\left(x\right) + \tan\left(x\right)$

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

Step-by-step solution

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

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

Step 1 — Sum/Difference Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}(u \pm v \pm \dots) = \frac{d}{dx}(u) \pm \frac{d}{dx}(v) \pm \dots$$
With:
  • $u = \sec\left(x\right)$
  • $v = \tan\left(x\right)$

Why: Differentiate every term separately and keep the plus and minus signs between them. Even a constant term first receives its own derivative symbol.

How does this rule work? Wikipedia
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sec\left(x\right) + \tan\left(x\right)\right]}\,}}}$$
✨ After: Sum/Difference Rule
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sec\left(x\right)\right]} + \frac{d}{dx}{\left[\tan\left(x\right)\right]}\,}}}$$
Step 2 — Derivative of the secant Differentiation rule
📖 Rule
$$\frac{d}{dx}\sec\left(x\right)=\sec\left(x\right)\,\tan\left(x\right)$$

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

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

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