Derivative of $x\,\sec\left(x\right)^{2}$
Differentiated with respect to x. Edit the function · Differentiation rules
Step-by-step solution
Input function
$$x\,\sec\left(x\right)^{2}$$
Start of the differentiation
$$\frac{d}{dx}{\left[x\,\sec\left(x\right)^{2}\right]}$$
Yellow: the expression being replaced. Green: its replacement.
Step 1 — Product Rule
Differentiation rule
📖 Rule
$$\frac{d}{dx}(u \cdot v)=u'v+uv'$$
With:
- $u = x$
- $v = \sec\left(x\right)^{2}$
Why: First, differentiate the first factor and multiply by the unchanged second factor. Add the unchanged first factor times the derivative of the second. Do not simply multiply the two derivatives.
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\,\sec\left(x\right)^{2}\right]}\,}}}$$
✨ After: Product Rule
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\right]}\,\sec\left(x\right)^{2} + x\,\frac{d}{dx}{\left[\sec\left(x\right)^{2}\right]}\,}}}$$
Step 2 — Derivative of a Variable
Differentiation rule
📖 Rule
$$\frac{d}{dx}(x)=1$$
Why: When x increases by one, the value of this term also increases by one. Its derivative is therefore one.
Changed part:
$\frac{d}{dx}{\left[x\right]} \;\Longrightarrow\; 1$
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\right]}\,}}}\,\sec\left(x\right)^{2} + x\,\frac{d}{dx}{\left[\sec\left(x\right)^{2}\right]}$$
✨ After: Derivative of a Variable
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}\,\sec\left(x\right)^{2} + x\,\frac{d}{dx}{\left[\sec\left(x\right)^{2}\right]}$$
Step 3 — Remove a factor of one
Simplification
📖 Rule
$$1\cdot a=a,\quad \frac{a}{1}=a$$
Why: Multiplication or division by one leaves the value unchanged.
Changed part:
$1\,\sec\left(x\right)^{2} \;\Longrightarrow\; \sec\left(x\right)^{2}$
🧮 Current expression
$$\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,\sec\left(x\right)^{2}\,}}} + x\,\frac{d}{dx}{\left[\sec\left(x\right)^{2}\right]}$$
✨ After: Remove a factor of one
$$\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\sec\left(x\right)^{2}\,}}} + x\,\frac{d}{dx}{\left[\sec\left(x\right)^{2}\right]}$$
Step 4 — Power Rule
Differentiation rule
📖 Rule
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}\,u'$$
With:
- $u = \sec\left(x\right)$
- $c = 2$
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:
$\frac{d}{dx}{\left[\sec\left(x\right)^{2}\right]} \;\Longrightarrow\; 2\,\sec\left(x\right)^{2 - 1}\,\frac{d}{dx}{\left[\sec\left(x\right)\right]}$
🧮 Current expression
$$\sec\left(x\right)^{2} + x\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sec\left(x\right)^{2}\right]}\,}}}$$
✨ After: Power Rule
$$\sec\left(x\right)^{2} + x \cdot \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,\sec\left(x\right)^{2 - 1}\,\frac{d}{dx}{\left[\sec\left(x\right)\right]}\,}}}$$
Step 5 — Evaluate numbers
Simplification
📖 Rule
Why: Evaluate only the highlighted numeric expression, preserving exact fractions and decimals.
Changed part:
$2 - 1 \;\Longrightarrow\; 1$
🧮 Current expression
$$\sec\left(x\right)^{2} + x \cdot 2\,\sec\left(x\right)^{\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 - 1\,}}}}\,\frac{d}{dx}{\left[\sec\left(x\right)\right]}$$
✨ After: Evaluate numbers
$$\sec\left(x\right)^{2} + x \cdot 2\,\sec\left(x\right)^{\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}\,\frac{d}{dx}{\left[\sec\left(x\right)\right]}$$
Step 6 — Power with exponent one
Simplification
📖 Rule
$$a^1=a$$
Why: An expression to the first power is the expression itself.
Changed part:
$\sec\left(x\right)^{1} \;\Longrightarrow\; \sec\left(x\right)$
🧮 Current expression
$$\sec\left(x\right)^{2} + x \cdot 2\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\sec\left(x\right)^{1}\,}}}\,\frac{d}{dx}{\left[\sec\left(x\right)\right]}$$
✨ After: Power with exponent one
$$\sec\left(x\right)^{2} + x \cdot 2\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\sec\left(x\right)\,}}}\,\frac{d}{dx}{\left[\sec\left(x\right)\right]}$$
Step 7 — 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:
$\frac{d}{dx}{\left[\sec\left(x\right)\right]} \;\Longrightarrow\; \sec\left(x\right)\,\tan\left(x\right)$
🧮 Current expression
$$\sec\left(x\right)^{2} + x \cdot 2\,\sec\left(x\right)\,\bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sec\left(x\right)\right]}\,}}}$$
✨ After: Derivative of the secant
$$\sec\left(x\right)^{2} + x \cdot 2\,\sec\left(x\right)\,\bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\sec\left(x\right)\,\tan\left(x\right)\,}}}$$
Step 8 — 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.
Changed part:
$x \cdot 2\,\sec\left(x\right)\,\sec\left(x\right)\,\tan\left(x\right) \;\Longrightarrow\; 2\,x\,\sec\left(x\right)\,\sec\left(x\right)\,\tan\left(x\right)$
🧮 Current expression
$$\sec\left(x\right)^{2} + \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,x \cdot 2\,\sec\left(x\right)\,\sec\left(x\right)\,\tan\left(x\right)\,}}}$$
✨ After: Order the factors
$$\sec\left(x\right)^{2} + \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x\,\sec\left(x\right)\,\sec\left(x\right)\,\tan\left(x\right)\,}}}$$
Step 9 — Write identical factors as a square
Simplification
📖 Rule
$$a\cdot a=a^2$$
Why: An expression multiplied by itself is its square.
Changed part:
$2\,x\,\sec\left(x\right)\,\sec\left(x\right)\,\tan\left(x\right) \;\Longrightarrow\; 2\,x\,\sec\left(x\right)^{2}\,\tan\left(x\right)$
🧮 Current expression
$$\sec\left(x\right)^{2} + \bbox[#fff1b8, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x\,\sec\left(x\right)\,\sec\left(x\right)\,\tan\left(x\right)\,}}}$$
✨ After: Write identical factors as a square
$$\sec\left(x\right)^{2} + \bbox[#d6f5df, padding:0.15em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x\,\sec\left(x\right)^{2}\,\tan\left(x\right)\,}}}$$
Final result
$$\sec\left(x\right)^{2} + 2\,x\,\sec\left(x\right)^{2}\,\tan\left(x\right)$$
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.
Cross-check with a computer algebra system
The derivative computed directly by the computer algebra system Maxima:
$$2\,x\,\sec\left(x\right)^{2}\,\tan\left(x\right) + \sec\left(x\right)^{2}$$
Step-by-step result ↔ Maxima: identical ✓