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To find the derivative of a polynomial function like $f(x) = x^2$, identify the exponent n and apply the power rule f'(x) = nx^{n-1}. Bring the exponent down as a coefficient and reduce it by one.

Conditions: n is a fixed real exponent.; Use $x>0$ as the common real domain; positive integer powers extend over all real x and negative integer powers exclude $x=0$.; The source presents $n\ne 0$; the constant $n=0$ case is treated separately.

Each of the seven adjusted orange controls encodes a discrete pair consisting of an input $x$ and an estimated slope $m$. Once these individual samples are deemed sufficiently accurate by the software, it automatically interpolates them to draw a continuous yellow parabola representing the full derivative function across the displayed domain.

Conditions: Enough accuracy has been achieved in estimating the individual slopes.; The underlying function remains the same displayed $f(x)=2x^3$ throughout.

Starting from $Av = \lambda v$, we rewrite the right side as $(\lambda I)v$ and move all terms to one side to get $(A - \lambda I)v = 0$. Since we seek non-zero solutions for $v$, the matrix $(A - \lambda I)$ must squash space into a lower dimension (have a non-trivial null space).

Conditions: $v$ is a non-zero eigenvector; $A$ is a square matrix; $I$ is the identity matrix

The indefinite integral of $x^2$ yields a family of functions $F(x) = \frac{x^3}{3} + C$. By defining the accumulation function $A(x)$ such that $A(0) = \int_0^0 t^2 dt = 0$, we impose an initial condition.

Conditions: Integrand is $f(t) = t^2$; Lower limit of integration is 0; Accumulation starts from zero

After determining the slope $m$, substitute it and the coordinates of a known point $(x_1, y_1)$ on the line into the point-slope formula $y - y_1 = m(x - x_1)$. The resulting equation is then algebraically simplified by distributing the slope and isolating $y$ to convert it into slope-intercept form $y = mx + b$.

Conditions: The slope $m$ of the line is already known.; At least one point $(x_1, y_1)$ that lies on the line is known.; The goal is to find the equation of the line.

For a fair six-sided die, the probability of rolling an even number is calculated using the classical probability formula. You count the number of favorable outcomes (even faces) and divide by the total number of possible outcomes (all faces).

Conditions: The die is fair (each face has equal probability).; The sample space is finite and consists of the numbers 1 through 6.; The event of interest is rolling an even number.

To calculate the probability of rolling an even number three times, first find the probability of rolling an even number on a single roll, which is $1/2$ for a fair six-sided die. Then, assuming the three rolls are mutually independent, multiply this single-roll probability by itself three times: $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$.

Conditions: The die is fair (six-sided, equally likely outcomes).; The three rolls are mutually independent.; The target event is rolling an even number on every one of the three rolls.

Euler's formula expresses the rotating complex exponential $e^{-2\pi i ft}$ as $\cos(2\pi ft) - i\sin(2\pi ft)$. When integrated against a signal $g(t)$, this separates the Fourier transform into two distinct real-valued integrals: the real part corresponds to the correlation with cosine, and the imaginary part corresponds to the correlation with negative sine.

Conditions: Using the convention where negative exponents denote clockwise rotation.; $g(t)$ is a real-valued function.