Derivative of $\sin\left(x\right)^{2} + \cos\left(x\right)^{2}$
Differentiated with respect to x. Edit the function · Differentiation rules
Step-by-step solution
Input function
$$\sin\left(x\right)^{2} + \cos\left(x\right)^{2}$$
Start of the differentiation
$$\frac{d}{dx}{\left[\sin\left(x\right)^{2} + \cos\left(x\right)^{2}\right]}$$
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:
- $u = \sin\left(x\right)^{2}$
- $v = \cos\left(x\right)^{2}$
Why: Differentiate every term separately and keep the plus and minus signs between them. Even a constant term first receives its own derivative symbol.
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)^{2} + \cos\left(x\right)^{2}\right]}\,}}}$$
✨ After: Sum/Difference Rule
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)^{2}\right]} + \frac{d}{dx}{\left[\cos\left(x\right)^{2}\right]}\,}}}$$
Step 2 — Power Rule
Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}\,u'$$
With:
- $u = \sin\left(x\right)$
- $c = 2$
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.
Changed part:
$\frac{d}{dx}{\left[\sin\left(x\right)^{2}\right]} \;\Longrightarrow\; 2\,\sin\left(x\right)^{2 - 1}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}$
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)^{2}\right]}\,}}} + \frac{d}{dx}{\left[\cos\left(x\right)^{2}\right]}$$
✨ After: Power Rule
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\sin\left(x\right)^{2 - 1}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}\,}}} + \frac{d}{dx}{\left[\cos\left(x\right)^{2}\right]}$$
Step 3 — Evaluate numbers
Simplification
📖 Rule
Why: Evaluate only the highlighted numeric expression, preserving exact fractions and decimals.
Changed part:
$2 - 1 \;\Longrightarrow\; 1$
🧮 Current expression
$$2\,\sin\left(x\right)^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 - 1\,}}}}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]} + \frac{d}{dx}{\left[\cos\left(x\right)^{2}\right]}$$
✨ After: Evaluate numbers
$$2\,\sin\left(x\right)^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]} + \frac{d}{dx}{\left[\cos\left(x\right)^{2}\right]}$$
Step 4 — Power with exponent one
Simplification
📖 Rule
$$a^1=a$$
Why: An expression to the first power is the expression itself.
Changed part:
$\sin\left(x\right)^{1} \;\Longrightarrow\; \sin\left(x\right)$
🧮 Current expression
$$2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\sin\left(x\right)^{1}\,}}}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]} + \frac{d}{dx}{\left[\cos\left(x\right)^{2}\right]}$$
✨ After: Power with exponent one
$$2\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\sin\left(x\right)\,}}}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]} + \frac{d}{dx}{\left[\cos\left(x\right)^{2}\right]}$$
Step 5 — Derivative of Sine
Differentiation rule
📖 Rule
$$\frac{d}{dx}\sin\left(x\right)=\cos\left(x\right)$$
Why: The argument is exactly x. Apply the known elementary derivative.
Changed part:
$\frac{d}{dx}{\left[\sin\left(x\right)\right]} \;\Longrightarrow\; \cos\left(x\right)$
🧮 Current expression
$$2\,\sin\left(x\right)\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}\,}}} + \frac{d}{dx}{\left[\cos\left(x\right)^{2}\right]}$$
✨ After: Derivative of Sine
$$2\,\sin\left(x\right)\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\cos\left(x\right)\,}}} + \frac{d}{dx}{\left[\cos\left(x\right)^{2}\right]}$$
Step 6 — 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.
Changed part:
$\frac{d}{dx}{\left[\cos\left(x\right)^{2}\right]} \;\Longrightarrow\; 2\,\cos\left(x\right)^{2 - 1}\,\frac{d}{dx}{\left[\cos\left(x\right)\right]}$
🧮 Current expression
$$2\,\sin\left(x\right)\,\cos\left(x\right) + \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\cos\left(x\right)^{2}\right]}\,}}}$$
✨ After: Power Rule
$$2\,\sin\left(x\right)\,\cos\left(x\right) + \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\cos\left(x\right)^{2 - 1}\,\frac{d}{dx}{\left[\cos\left(x\right)\right]}\,}}}$$
Step 7 — Evaluate numbers
Simplification
📖 Rule
Why: Evaluate only the highlighted numeric expression, preserving exact fractions and decimals.
Changed part:
$2 - 1 \;\Longrightarrow\; 1$
🧮 Current expression
$$2\,\sin\left(x\right)\,\cos\left(x\right) + 2\,\cos\left(x\right)^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 - 1\,}}}}\,\frac{d}{dx}{\left[\cos\left(x\right)\right]}$$
✨ After: Evaluate numbers
$$2\,\sin\left(x\right)\,\cos\left(x\right) + 2\,\cos\left(x\right)^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}\,\frac{d}{dx}{\left[\cos\left(x\right)\right]}$$
Step 8 — Power with exponent one
Simplification
📖 Rule
$$a^1=a$$
Why: An expression to the first power is the expression itself.
Changed part:
$\cos\left(x\right)^{1} \;\Longrightarrow\; \cos\left(x\right)$
🧮 Current expression
$$2\,\sin\left(x\right)\,\cos\left(x\right) + 2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\cos\left(x\right)^{1}\,}}}\,\frac{d}{dx}{\left[\cos\left(x\right)\right]}$$
✨ After: Power with exponent one
$$2\,\sin\left(x\right)\,\cos\left(x\right) + 2\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\cos\left(x\right)\,}}}\,\frac{d}{dx}{\left[\cos\left(x\right)\right]}$$
Step 9 — 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:
$\frac{d}{dx}{\left[\cos\left(x\right)\right]} \;\Longrightarrow\; -\sin\left(x\right)$
🧮 Current expression
$$2\,\sin\left(x\right)\,\cos\left(x\right) + 2\,\cos\left(x\right)\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\cos\left(x\right)\right]}\,}}}$$
✨ After: Derivative of Cosine
$$2\,\sin\left(x\right)\,\cos\left(x\right) + 2\,\cos\left(x\right)\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\left(-\sin\left(x\right)\right)\,}}}$$
Step 10 — 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.
Changed part:
$2\,\cos\left(x\right)\,\left(-\sin\left(x\right)\right) \;\Longrightarrow\; -2\,\cos\left(x\right)\,\sin\left(x\right)$
🧮 Current expression
$$2\,\sin\left(x\right)\,\cos\left(x\right) + \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\cos\left(x\right)\,\left(-\sin\left(x\right)\right)\,}}}$$
✨ After: Determine the sign of the product
$$2\,\sin\left(x\right)\,\cos\left(x\right) + \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\left(-2\,\cos\left(x\right)\,\sin\left(x\right)\right)\,}}}$$
Step 11 — Combine plus and minus
Simplification
📖 Rule
$$a+(-b)=a-b$$
Why: Adding a negative expression is the same as subtracting the positive expression.
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\sin\left(x\right)\,\cos\left(x\right) + \left(-2\,\cos\left(x\right)\,\sin\left(x\right)\right)\,}}}$$
✨ After: Combine plus and minus
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\sin\left(x\right)\,\cos\left(x\right) - 2\,\cos\left(x\right)\,\sin\left(x\right)\,}}}$$
Step 12 — Opposite terms cancel
Simplification
📖 Rule
$$u-u=0$$
Why: The same term appears once with plus and once with minus. Together they give zero, so both are removed.
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\sin\left(x\right)\,\cos\left(x\right) - 2\,\cos\left(x\right)\,\sin\left(x\right)\,}}}$$
✨ After: Opposite terms cancel
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,0\,}}}$$
Final result
Cross-check with a computer algebra system
The derivative computed directly by the computer algebra system Maxima:
$$0$$
Step-by-step result ↔ Maxima: identical ✓