Question 1

What is a derivative in calculus?

Accepted Answer

A derivative f'(x) is the instantaneous rate of change of a function — the slope of the tangent line to the curve at each point. Formally, f'(x) = lim[h→0] (f(x+h)−f(x))/h. Key rules: power rule d/dx[xⁿ]=nxⁿ⁻¹, chain rule d/dx[f(g(x))]=f'(g(x))·g'(x), product rule d/dx[fg]=f'g+fg', quotient rule d/dx[f/g]=(f'g−fg')/g².

Question 2

What is an integral in calculus?

Accepted Answer

An integral is the reverse of differentiation. The indefinite integral ∫f(x)dx = F(x)+C finds all antiderivatives. The definite integral ∫[a→b]f(x)dx = F(b)−F(a) computes the signed area under the curve from a to b. Key rules: power rule ∫xⁿdx = xⁿ⁺¹/(n+1)+C (n≠−1), ∫eˣdx = eˣ+C, ∫sin x dx = −cos x + C.

Question 3

What is the Fundamental Theorem of Calculus?

Accepted Answer

The FTC has two parts: Part 1 — d/dx[∫[a→x]f(t)dt] = f(x): differentiating an integral returns the original function. Part 2 — ∫[a→b]f(x)dx = F(b)−F(a): to evaluate a definite integral, find any antiderivative and evaluate at the bounds. This connects differentiation and integration as inverse operations.

Question 4

What is the chain rule?

Accepted Answer

The chain rule differentiates composite functions: d/dx[f(g(x))] = f'(g(x))·g'(x). Identify the outer and inner functions, differentiate each, then multiply. Example: d/dx[sin(3x²)] = cos(3x²)·6x = 6x·cos(3x²). The chain rule is the most frequently used differentiation rule in practice.

Question 5

What is a limit in calculus?

Accepted Answer

lim[x→a]f(x) = L means f(x) approaches L as x approaches a. For the limit to exist, the left-hand and right-hand limits must both equal L. Common techniques: direct substitution, factoring to remove discontinuities, and L'Hôpital's rule for 0/0 or ∞/∞ forms. Derivatives and integrals are both defined as limits.

Question 6

What is a Riemann sum?

Accepted Answer

A Riemann sum approximates a definite integral by dividing the area under a curve into n rectangles (or trapezoids) and summing their areas. Left, right, and midpoint sums use rectangle heights; the trapezoidal rule uses trapezoids. As n→∞, the sum converges to the exact integral. The midpoint rule and trapezoidal rule both achieve O(1/n²) accuracy.

Question 7

What is implicit differentiation?

Accepted Answer

Implicit differentiation finds dy/dx when y is not explicitly solved for x. Differentiate both sides of the equation with respect to x, treating y as a function of x — each y term gains a factor of dy/dx via the chain rule. Then solve for dy/dx. Example: x²+y²=25 → 2x+2y(dy/dx)=0 → dy/dx=−x/y.

f(x)	f'(x)	f(x)	f'(x)
c	0	sin x	cos x
xⁿ	nxⁿ⁻¹	cos x	−sin x
eˣ	eˣ	tan x	sec²x
aˣ	aˣ ln a	cot x	−csc²x
ln x	1/x	sec x	sec x tan x
logₐx	1/(x ln a)	arcsin x	1/√(1−x²)
√x	1/(2√x)	arctan x	1/(1+x²)

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