Derivative of sin²(x)
sin²(x) means (sin x)². Use the power rule on the outside, multiply by the derivative of sine, and the double-angle identity condenses the result.
Step-by-step solution
sin²(x)=(sin x)². Let u=sin(x); the outer function is u². In the calculator, type sin(x)^2.
The outer derivative is 2u; the inner derivative is cos(x).
f′(x)=2sin(x)cos(x). By the double-angle identity this equals sin(2x).
Domain and graph behavior
Both the function and derivative are defined for every real x. Because sin²(x)=(1−cos 2x)/2, the graph repeats every π rather than every 2π, and its values stay in [0, 1]. The derivative sin(2x) is zero at x=kπ/2: minima with value 0 occur at x=kπ, and maxima with value 1 at x=π/2+kπ.
Second derivative
f″(x)=d/dx sin(2x)=2cos(2x), which also equals 2cos²(x)−2sin²(x). It is zero at x=π/4+kπ/2, where the function equals 1/2; those are the inflection points between each valley and peak.
Common mistake
Do not confuse sin²(x) with sin(x²). The first squares the output of sine; the second squares the input, and its derivative is 2x·cos(x²). Also avoid stopping at 2sin(x)—the factor cos(x) from the inner function is required.
Practice problems
Differentiate sin²(3x)
Answer: 2sin(3x)·cos(3x)·3=3sin(6x): there are two inner layers, sine and 3x.
Differentiate sin³(x)
Answer: 3sin²(x)cos(x): the power rule lowers the exponent by one, then multiply by cos(x).
Check with the power-reduction identity
Rewrite sin²(x)=(1−cos 2x)/2 before differentiating. The constant 1/2 contributes nothing, and −cos(2x)/2 differentiates to (2sin 2x)/2=sin(2x). Reaching sin(2x) without the chain rule on sine confirms the answer.
Check with the product rule
Treat sin²(x) as sin(x)·sin(x). The product rule gives cos(x)sin(x)+sin(x)cos(x)=2sin(x)cos(x). This is the same result, which shows why the power rule's factor 2 appears: two identical factors each contribute one copy.
Connection with cos²(x)
Since sin²(x)+cos²(x)=1 for every x, the derivatives must add to zero. Indeed, d/dx cos²(x)=−sin(2x). Whatever sin²(x) gains, cos²(x) loses at the same instant. See the cos²(x) solution for the mirror-image calculation.
A numerical check
At x=π/4 the formula gives sin(π/2)=1. At x=π/6 it gives sin(π/3)=√3/2≈0.8660. Centered secants with h=0.0001 reproduce both values to at least six decimal places.
More worked variations
x·sin²(x): product rule gives sin²(x)+x·sin(2x).
sinⁿ(x): n·sinⁿ⁻¹(x)·cos(x) for any constant n.
sin²(x)+cos²(x): sin(2x)−sin(2x)=0, as expected for the constant 1.
Questions students ask
Should I write 2sin(x)cos(x) or sin(2x)?
Both are correct and equal for every x. sin(2x) is more compact; 2sin(x)cos(x) shows the chain rule more clearly. Use whichever form your course expects.
Does the superscript in sin²(x) work like the one in sin⁻¹(x)?
No. The superscript 2 means squaring the output. The superscript −1 is the exception: sin⁻¹(x) usually means arcsin(x), not 1/sin(x).
Related derivatives
Review the chain rule and the trigonometric derivatives guide, or read our verification method and limitations.