Derivative Calculator

Derivative of $\frac{1}{\cos\left(x\right)^{2}}$

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

Step-by-step solution

Input function
$$\frac{1}{\cos\left(x\right)^{2}}$$
Start of the differentiation
$$\frac{d}{dx}{\left[\frac{1}{\cos\left(x\right)^{2}}\right]}$$

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

Step 1 — Write the reciprocal as a negative power Rewriting
📖 Rule
$$\frac{c}{v}=c\cdot v^{-1},\quad \frac{c}{v^k}=c\cdot v^{-k}$$

Why: Write the denominator as a power with a negative exponent, then apply the factor and power rules. The original denominator must still be nonzero.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\frac{1}{\cos\left(x\right)^{2}}\right]}\,}}}$$
✨ After: Write the reciprocal as a negative power
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\cos\left(x\right)^{-2}\right]}\,}}}$$
Step 2 — Power Rule Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}\,u'$$
With:
  • $u = \cos\left(x\right)$
  • $c = -2$

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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\cos\left(x\right)^{-2}\right]}\,}}}$$
✨ After: Power Rule
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-2\,\cos\left(x\right)^{-2 - 1}\,\frac{d}{dx}{\left[\cos\left(x\right)\right]}\,}}}$$
Step 3 — Evaluate numbers Simplification
📖 Rule

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

Changed part: $-2 - 1 \;\Longrightarrow\; -3$
🧮 Current expression
$$-2\,\cos\left(x\right)^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-2 - 1\,}}}}\,\frac{d}{dx}{\left[\cos\left(x\right)\right]}$$
✨ After: Evaluate numbers
$$-2\,\cos\left(x\right)^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-3\,}}}}\,\frac{d}{dx}{\left[\cos\left(x\right)\right]}$$
Step 4 — Derivative of Cosine Differentiation rule
📖 Rule
$$\frac{d}{dx}\cos\left(x\right)=-\sin\left(x\right)$$

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

Changed part: $\frac{d}{dx}{\left[\cos\left(x\right)\right]} \;\Longrightarrow\; -\sin\left(x\right)$
How does this rule work? Wikipedia
🧮 Current expression
$$-2\,\cos\left(x\right)^{-3}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\cos\left(x\right)\right]}\,}}}$$
✨ After: Derivative of Cosine
$$-2\,\cos\left(x\right)^{-3}\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\left(-\sin\left(x\right)\right)\,}}}$$
Step 5 — Determine the sign of the product Simplification
📖 Rule
$$a\cdot(-b)=-(a\cdot b),\quad (-a)\cdot(-b)=a\cdot b$$

Why: Move the minus signs of the factors in front of the product. An even number of minus signs gives plus, an odd number gives minus.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-2\,\cos\left(x\right)^{-3}\,\left(-\sin\left(x\right)\right)\,}}}$$
✨ After: Determine the sign of the product
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\cos\left(x\right)^{-3}\,\sin\left(x\right)\,}}}$$
Step 6 — Write negative exponents as a fraction Simplification
📖 Rule
$$a^{-n}=\frac{1}{a^n}$$

Why: A power with a negative exponent is the reciprocal of the power with the positive exponent. The result then has no negative exponents.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\cos\left(x\right)^{-3}\,\sin\left(x\right)\,}}}$$
✨ After: Write negative exponents as a fraction
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{2\,\sin\left(x\right)}{\cos\left(x\right)^{3}}\,}}}$$
Final result
$$\frac{2\,\sin\left(x\right)}{\cos\left(x\right)^{3}}$$
Valid where:
  • $\cos\left(x\right)^{2}\ne0$ – The original denominator must be nonzero.

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:

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