Derivative Calculator

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

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

Step-by-step solution

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

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

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

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

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

Why: Differentiate the outer function while keeping its inner argument u. Multiply by u' and evaluate that inner derivative in its own step.

Changed part: $\frac{d}{dx}{\left[\sin\left(x^{2}\right)\right]} \;\Longrightarrow\; \cos\left(x^{2}\right)\,\frac{d}{dx}{\left[x^{2}\right]}$
How does this rule work? Wikipedia
🧮 Current expression
$$-1\,\sin\left(x^{2}\right)^{-2}\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x^{2}\right)\right]}\,}}}$$
✨ After: Derivative of Sine (Chain Rule)
$$-1\,\sin\left(x^{2}\right)^{-2}\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\cos\left(x^{2}\right)\,\frac{d}{dx}{\left[x^{2}\right]}\,}}}$$
Step 5 — Power Rule (special case) Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}$$
With:
  • $u = x$
  • $c = 2$

Why: Bring the current exponent down as a multiplier and subtract one from the exponent. Evaluate that small calculation in a separate step.

Changed part: $\frac{d}{dx}{\left[x^{2}\right]} \;\Longrightarrow\; 2\,x^{2 - 1}$
How does this rule work? Wikipedia
🧮 Current expression
$$-1\,\sin\left(x^{2}\right)^{-2}\,\cos\left(x^{2}\right)\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2}\right]}\,}}}$$
✨ After: Power Rule (special case)
$$-1\,\sin\left(x^{2}\right)^{-2}\,\cos\left(x^{2}\right) \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x^{2 - 1}\,}}}$$
Step 6 — Evaluate numbers Simplification
📖 Rule

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

Changed part: $2 - 1 \;\Longrightarrow\; 1$
🧮 Current expression
$$-1\,\sin\left(x^{2}\right)^{-2}\,\cos\left(x^{2}\right) \cdot 2\,x^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 - 1\,}}}}$$
✨ After: Evaluate numbers
$$-1\,\sin\left(x^{2}\right)^{-2}\,\cos\left(x^{2}\right) \cdot 2\,x^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}$$
Step 7 — Power with exponent one Simplification
📖 Rule
$$a^1=a$$

Why: An expression to the first power is the expression itself.

Changed part: $x^{1} \;\Longrightarrow\; x$
🧮 Current expression
$$-1\,\sin\left(x^{2}\right)^{-2}\,\cos\left(x^{2}\right) \cdot 2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x^{1}\,}}}$$
✨ After: Power with exponent one
$$-1\,\sin\left(x^{2}\right)^{-2}\,\cos\left(x^{2}\right) \cdot 2\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x\,}}}$$
Step 8 — Write the factor −1 as a minus sign Simplification
📖 Rule
$$(-1)\cdot a=-a$$

Why: Multiplying by −1 gives the negative. Write just a minus sign instead.

🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-1\,\sin\left(x^{2}\right)^{-2}\,\cos\left(x^{2}\right) \cdot 2\,x\,}}}$$
✨ After: Write the factor −1 as a minus sign
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\sin\left(x^{2}\right)^{-2}\,\cos\left(x^{2}\right) \cdot 2\,x\,}}}$$
Step 9 — 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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\sin\left(x^{2}\right)^{-2}\,\cos\left(x^{2}\right) \cdot 2\,x\,}}}$$
✨ After: Order the factors
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-2\,x\,\sin\left(x^{2}\right)^{-2}\,\cos\left(x^{2}\right)\,}}}$$
Step 10 — 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
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-2\,x\,\sin\left(x^{2}\right)^{-2}\,\cos\left(x^{2}\right)\,}}}$$
✨ After: Write negative exponents as a fraction
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\frac{2\,x\,\cos\left(x^{2}\right)}{\sin\left(x^{2}\right)^{2}}\,}}}$$
Final result
$$-\frac{2\,x\,\cos\left(x^{2}\right)}{\sin\left(x^{2}\right)^{2}}$$
Valid where:
  • $\sin\left(x^{2}\right)\ne0$ – 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:

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