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Proof that the derivative of a constant function is zero

This proof is short, but it matters because it shows the derivative definition already knows that a function with no change has zero slope. Nothing extra has to be added as a rule.

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Proof that $\frac{d}{d x} x^{n} = n x^{n - 1}$ for positive integers

This proof shows where the power rule comes from rather than treating it as a black box. The binomial theorem isolates the only term that survives after dividing by $h$ and taking the limit.

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Proof of the chain rule

The chain rule says the derivative of $f (g (x))$ is $f^{'} (g (x)) \cdot g^{'} (x)$ . The proof uses a continuous extension of the difference quotient to handle all cases cleanly.

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Proof that a differentiable function is continuous

This proof shows why differentiability is a stronger condition than continuity. The difference $f (a + h) - f (a)$ factors into a difference quotient times $h$ , and that product is forced to zero.

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Worked Examples

Common examples from different branches of math, each written with clear step-by-step instructions.

Differentiate a polynomial from first principles

This worked example shows how the derivative definition creates an algebra problem first and a limit problem second. The simplification step is the point of the exercise.

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Evaluate an integral by integration by parts

A simple example like $\int x e^{x} d x$ is useful because it makes the choice of $u$ and $d v$ visible without extra algebra getting in the way.

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Evaluate an integral by substitution

Substitution is the chain rule in reverse. In $\int 2 x cos (x^{2}) d x$ , the factor $2 x$ matches the derivative of the inner expression $x^{2}$ , so the structure is visible without extra algebra.

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Solve an optimization problem for the maximum area rectangle under a constraint

With perimeter fixed at $24$ , the area becomes the quadratic $A (x) = x (12 - x)$ . That makes the optimization workflow concrete: use the constraint to reduce to one variable, then locate the peak of the area function.

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Compare left Riemann sums, right Riemann sums, and midpoint sums for the same function

For $f (x) = x^{2}$ on $[0, 1]$ , left, right, and midpoint sums built from the same partition give noticeably different totals. The point of the page is to compare those behaviors directly rather than treating the formulas as interchangeable.

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Find and classify critical points of a quartic function

This worked example shows the standard calculus workflow: differentiate, solve $f^{'} (x) = 0$ , then use the second derivative and the graph to decide which critical points are local maxima or minima.

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Test convergence of an improper integral

An improper integral is decided by a limit, not by the antiderivative alone. This example tests $\int_{1}^{\infty} \frac{1}{x ^{2}} d x$ and compares it with the divergent harmonic-tail integral.

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Find a Taylor polynomial and estimate the error

This example builds the degree-5 Taylor polynomial of $sin (x)$ centred at $0$ , evaluates it at $x = 0.5$ , and confirms that the actual error stays inside the bound given by Taylor's theorem.

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Explanations

Concept pages that build intuition first and then connect it to formal notation and MCPCalc tools.

What a derivative means geometrically

The derivative is the slope of the tangent line, but that phrase only becomes useful when you compare the curve to nearby secant lines and local linear approximations.

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Why continuity and differentiability are different concepts

Differentiability is stronger than continuity. A graph can be unbroken at a point and still fail to have a derivative there if the local shape has a corner, cusp, or vertical tangent.

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What a Taylor polynomial is approximating

A Taylor polynomial is not trying to copy a function everywhere. It matches the function and several of its derivatives at one chosen center, so it is designed to be locally accurate near that point.

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