Math · Calculus I · Concept
The power rule and basic derivative rules
To differentiate a polynomial or a sum of powers, use three rules: the power rule d/dx xⁿ = nxⁿ⁻¹, the constant multiple rule (a constant factor stays) and the sum rule (differentiate term by term). Rewrite roots and fractions as powers first.
The power rule
For a power of x, bring the exponent down as a factor and lower the exponent by one. The rule holds for every real exponent n wherever xⁿ is defined and differentiable.
Why does the exponent come down?
For a positive integer n, expanding (x + h)ⁿ gives xⁿ + nxⁿ⁻¹h plus terms that contain h² or higher powers of h. After subtracting xⁿ and dividing by h, every term except nxⁿ⁻¹ still contains h and vanishes as h → 0.
Constants and sums
A constant factor stays in front: the derivative of c·f(x) is c·f′(x). A sum or difference is differentiated term by term. Together these rules let you differentiate a polynomial one term at a time.
A constant has derivative zero
The graph of y = c is horizontal, so its slope is 0 everywhere. Adding a constant to a function shifts its graph up or down without changing any of its slopes.
Rewrite roots and fractions as powers
The power rule needs the form xⁿ. Write √x as x^(1/2), a cube root as x^(1/3) and 1/x² as x^(−2), differentiate, then write the answer in whichever form is clearer.
Evaluate the derivative to get a slope
f′(x) is a new function. Substituting a number gives the slope of the tangent line at that input. For f(x) = x³ − 6x² + 9x, f′(x) = 3x² − 12x + 9 and f′(2) = −3, so the graph is falling at x = 2.
Common mistakes
- Keeping a constant term: the derivative of −2 is 0.
- Subtracting 1 from a negative exponent the wrong way: the derivative of x⁻² is −2x⁻³, not −2x⁻¹.
- Using the power rule on a product or quotient: x²(x + 1) needs the product rule or expanding first.
- Using the power rule on an exponential: 2ˣ and eˣ are not powers of x.
Key terms
- Power rule
- d/dx xⁿ = n·xⁿ⁻¹: bring down the exponent and lower it by one. It works for any real exponent n wherever the power is defined.
- Constant multiple rule
- The derivative of a constant times a function is the constant times the derivative: (c·f)′ = c·f′. A constant factor stays in front while the function is differentiated.
- Sum rule
- The derivative of a sum or difference is the sum or difference of the derivatives, so a polynomial is differentiated one term at a time.
- Derivative
- The instantaneous rate of change of a function: the limit of the average rate of change as the step shrinks to zero, when that limit exists. On a graph it is the slope of the tangent line.
- Polynomial
- An expression made by adding terms, each a number times whole-number powers of the variables, such as 3x² − 5x + 2. An expression with a variable in a denominator, under a root or inside a function like sin is not a polynomial.
Work through an example
Differentiate f(x) = 4√x − 3/x² + 5, then find the slope of its graph at x = 4.
Use the power rule on roots and fractions →Sources and scope
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