The Power Rule
The power rule is the most frequently used derivative rule in all of calculus.
f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. This works for any real number exponent n — positive, negative, or fractional.d/dx [x⁵] = 5x⁴.
d/dx [x⁻²] = −2x⁻³ = −2/x³.
d/dx [x^(1/2)] = (1/2)x^(−1/2) = 1/(2√x).
d/dx [x] = 1 (since x = x¹).
d/dx [constant] = 0 (since a constant = cx⁰, but more fundamentally because constants do not change).
When a constant multiplies a power, it just carries through: d/dx [7x³] = 21x².
The power rule also applies to terms like 1/x² = x⁻² and √x = x^(1/2). Getting comfortable rewriting expressions using exponents before differentiating is an essential skill.
Once the power rule is second nature, you can differentiate any polynomial term by term in seconds. This single rule handles roughly half of all the derivatives you will encounter in a first calculus course.
Sum, Difference, and Constant Multiple Rules
These rules let you break complex functions into simpler pieces.
d/dx [f(x) + g(x)] = f'(x) + g'(x).d/dx [f(x) − g(x)] = f'(x) − g'(x).d/dx [c · f(x)] = c · f'(x).These rules mean you can differentiate any polynomial term by term.
For example: d/dx [3x⁴ − 5x² + 2x − 7] = 12x³ − 10x + 2.
Each term is differentiated independently using the power rule, and the constant −7 vanishes.
For a more complex function like f(x) = 4√x + 3/x², rewrite as 4x^(1/2) + 3x⁻², then differentiate:
f'(x) = 4 · (1/2)x^(−1/2) + 3(−2)x⁻³ = 2/√x − 6/x³These three rules are so natural that experienced calculus students apply them without thinking. Together with the power rule, they handle all polynomial and many algebraic functions. The key habit is to always rewrite roots and fractions as power expressions before differentiating.
The Product Rule
When two functions are multiplied together, you need the product rule.
d/dx [f(x) · g(x)] = f'(x) · g(x) + f(x) · g'(x). A common mnemonic is "derivative of the first times the second, plus the first times the derivative of the second."d/dx [x² eˣ] = 2x eˣ + x² eˣ = eˣ(2x + x²) = x eˣ(2 + x)d/dx [x ln(x)] = 1 · ln(x) + x · (1/x) = ln(x) + 1d/dx [sin(x) cos(x)] = cos(x) cos(x) + sin(x)(−sin(x)) = cos²(x) − sin²(x) = cos(2x)A critical warning: the derivative of a product is not the product of the derivatives. That is, (fg)' ≠ f' · g'. This is one of the most common calculus mistakes.
The product rule extends to three or more factors: (fgh)' = f'gh + fg'h + fgh', though this situation is less common. Use the product rule whenever you see two non-constant functions multiplied at the outermost level of an expression.
The Quotient Rule
When one function is divided by another, use the quotient rule.
d/dx [f(x)/g(x)] = [f'(x) · g(x) − f(x) · g'(x)] / [g(x)]². The mnemonic "low d-high minus high d-low, all over the square of what's below" helps many students remember the formula.d/dx [x/(x² + 1)].
Let f = x, g = x² + 1. Then f' = 1, g' = 2x.
[1(x² + 1) − x(2x)] / (x² + 1)² = (x² + 1 − 2x²) / (x² + 1)² = (1 − x²) / (x² + 1)²d/dx [tan(x)] = d/dx [sin(x)/cos(x)].
Using the quotient rule: [cos(x) cos(x) − sin(x)(−sin(x))] / cos²(x) = (cos²(x) + sin²(x)) / cos²(x) = 1/cos²(x) = sec²(x).
sec²(x)This is how the derivative of tangent is derived.
Tip: You can often avoid the quotient rule by rewriting the expression. For instance, 1/x² = x⁻², so use the power rule instead: d/dx [x⁻²] = −2x⁻³. Solver AI will show you both approaches when applicable.
Derivatives of Trig, Exponential, and Log Functions
Beyond polynomial functions, you need to know the derivatives of the standard transcendental functions.
Trigonometric
d/dx [sin(x)] = cos(x). d/dx [cos(x)] = −sin(x). d/dx [tan(x)] = sec²(x). d/dx [sec(x)] = sec(x)tan(x). d/dx [csc(x)] = −csc(x)cot(x). d/dx [cot(x)] = −csc²(x).Exponential
d/dx [eˣ] = eˣ — the exponential function is its own derivative, one of its most remarkable properties. d/dx [aˣ] = aˣ ln(a) for any positive constant a.Logarithmic
d/dx [ln(x)] = 1/x. d/dx [logₐ(x)] = 1/(x ln a).Inverse trig
d/dx [arcsin(x)] = 1/√(1 − x²). d/dx [arctan(x)] = 1/(1 + x²). d/dx [arcsec(x)] = 1/(|x|√(x² − 1)).Memorize these derivatives — they are the building blocks that, combined with the product, quotient, and chain rules, let you differentiate virtually any function. A flashcard deck of these formulas is one of the most efficient study investments for calculus.
Derivative Formula Cheat Sheet
Here is a consolidated reference of all the derivative formulas you need, organized for quick lookup.
Basic
d/dx [c] = 0, d/dx [xⁿ] = nxⁿ⁻¹, d/dx [cf(x)] = cf'(x).Arithmetic
(f ± g)' = f' ± g', (fg)' = f'g + fg', (f/g)' = (f'g − fg')/g².Chain rule
d/dx [f(g(x))] = f'(g(x)) · g'(x).Trig
(sin x)' = cos x, (cos x)' = −sin x, (tan x)' = sec²x, (cot x)' = −csc²x, (sec x)' = sec x tan x, (csc x)' = −csc x cot x.Exponential/Log
(eˣ)' = eˣ, (aˣ)' = aˣ ln a, (ln x)' = 1/x, (logₐ x)' = 1/(x ln a).Inverse trig
(arcsin x)' = 1/√(1−x²), (arccos x)' = −1/√(1−x²), (arctan x)' = 1/(1+x²).Print this page or save it to your phone as a study reference. When practicing, try to recall the formula before looking it up. With Solver AI, you can verify any derivative instantly — type in the function and compare your hand-calculated result against the AI's step-by-step solution.