What is a Derivative?
A derivative measures how fast a function's output changes as its input changes — the instantaneous rate of change or slope of the tangent line at a point.
The derivative of f(x) is f'(x) = lim(h→0) [f(x+h)−f(x)]/h. For power functions, the power rule gives d/dx xⁿ = n·x^(n−1).
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Step-by-step worked examples
Find the derivative of f(x) = x⁴.
Apply power rule: d/dx xⁿ = n·x^(n−1) n=4 → f'(x) = 4x³
Find f'(2) for f(x) = x³.
f'(x) = 3x² f'(2) = 3(2²) = 3·4 = 12
Find the derivative of f(x) = 5x² + 3x.
Differentiate term by term. d/dx[5x²] = 10x d/dx[3x] = 3 f'(x) = 10x + 3
Flashcards
Quick quiz
Q1.Derivative of x⁵?
Q2.Derivative of a constant like 7?
Q3.f(x)=x³, f'(2)=?
Q4.Derivative of 4x² + 2x?
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Common mistakes
Forgetting to subtract 1 from the exponent. — Correct: Power rule: bring down n, then reduce the exponent by 1 → n·x^(n−1).
Treating the derivative of a constant as the constant itself. — Correct: Constants have zero rate of change, so their derivative is always 0.
Not differentiating each term separately in a sum. — Correct: Differentiate term-by-term: d/dx[f+g] = f'+g'.
Confusing f(x) with f'(x). — Correct: f(x) is the function's value; f'(x) is its rate of change/slope.
FAQ
What is a derivative?
A derivative is the instantaneous rate of change of a function — the slope of the tangent line at a given point.
What is the derivative formula (power rule)?
The power rule states d/dx [xⁿ] = n·x^(n−1), used to differentiate any power function.
What are examples of derivatives?
The derivative of x⁴ is 4x³, and the derivative of 5x²+3x is 10x+3, found by applying the power rule term by term.
How to calculate a derivative?
Apply the power rule to each term: bring the exponent down as a coefficient, then reduce the exponent by 1.