Mathematical Guide
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Calculus Derivative Rules & Step-by-Step Examples
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By CalcSolver Editorial Team
Calculus Derivative Rules & Step-by-Step Examples
A derivative measures how fast a function's output changes as its input changes — the slope of the tangent line at a given point. This cheat sheet collects every core derivative rule in one place, with plain-language explanations and copyable worked examples you can follow line by line. Use the quick reference table below to jump straight to the rule you need, or scroll through for full explanations.
Quick Reference: All Derivative Rules
| Rule | Formula | When to Use It |
|---|---|---|
| Constant Rule | d/dx[c] = 0 | Derivative of any constant |
| Power Rule | d/dx[x^n] = n·x^(n-1) | Any term with x raised to a power |
| Constant Multiple Rule | d/dx[c·f(x)] = c·f'(x) | A function scaled by a constant |
| Sum/Difference Rule | d/dx[f(x) ± g(x)] = f'(x) ± g'(x) | Functions added or subtracted |
| Product Rule | d/dx[f(x)·g(x)] = f'(x)g(x) + f(x)g'(x) | Two functions multiplied together |
| Quotient Rule | d/dx[f(x)/g(x)] = [f'(x)g(x) − f(x)g'(x)] / [g(x)]² | One function divided by another |
| Chain Rule | d/dx[f(g(x))] = f'(g(x))·g'(x) | A function nested inside another function |
Basic Derivative Rules
These five rules handle the majority of derivatives you'll encounter in an introductory calculus course. Master these before moving on to trig, exponential, and logarithmic derivatives.
1. Constant Rule
The derivative of any constant is zero, because a constant never changes — it has zero slope everywhere. Example: if f(x) = 12, then f'(x) = 0.
2. Power Rule
For any term x raised to a power n, bring the exponent down as a coefficient and reduce the exponent by 1: d/dx[x^n] = n·x^(n-1). This works for positive, negative, and fractional exponents alike.
Worked Example: Power Rule
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Function: f(x) = x^5
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Step 1 — Bring the exponent down as a coefficient: 5·x^(5-1)
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Step 2 — Subtract 1 from the exponent: 5·x^4
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Result: f'(x) = 5x^4
Worked Example: Power Rule with a Negative Exponent
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Function: f(x) = 1/x^2, rewritten as f(x) = x^(-2)
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Step 1 — Apply the power rule: -2·x^(-2-1)
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Step 2 — Simplify the exponent: -2·x^(-3)
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Result: f'(x) = -2/x^3
3. Constant Multiple Rule
A constant multiplying a function passes straight through the derivative unchanged: d/dx[c·f(x)] = c·f'(x). Example: if f(x) = 7x^3, then f'(x) = 7·(3x^2) = 21x^2.
4. Sum and Difference Rule
Differentiate term by term. The derivative of a sum or difference of functions is the sum or difference of their derivatives.
Worked Example: Sum/Difference Rule
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Function: f(x) = 4x^3 − 6x^2 + 9x − 5
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Step 1 — Differentiate each term separately using the power rule
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d/dx[4x^3] = 12x^2
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d/dx[-6x^2] = -12x
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d/dx[9x] = 9
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d/dx[-5] = 0
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Result: f'(x) = 12x^2 − 12x + 9
5. Product Rule
When two functions are multiplied together, you cannot just multiply their derivatives. Instead: d/dx[f(x)·g(x)] = f'(x)g(x) + f(x)g'(x) — derivative of the first times the second, plus the first times the derivative of the second.
Worked Example: Product Rule
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Function: h(x) = (x^2 + 1)(x^3 − 4x)
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Step 1 — Let f(x) = x^2 + 1, so f'(x) = 2x
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Step 2 — Let g(x) = x^3 − 4x, so g'(x) = 3x^2 − 4
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Step 3 — Apply the product rule: f'(x)g(x) + f(x)g'(x)
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= (2x)(x^3 − 4x) + (x^2 + 1)(3x^2 − 4)
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Step 4 — Expand: (2x^4 − 8x^2) + (3x^4 − 4x^2 + 3x^2 − 4)
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Result: h'(x) = 5x^4 − 9x^2 − 4
6. Quotient Rule
For one function divided by another: d/dx[f(x)/g(x)] = [f'(x)g(x) − f(x)g'(x)] / [g(x)]². A common memory trick is 'low d-high minus high d-low, over low-low': (g·f' − f·g') / g².
Worked Example: Quotient Rule
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Function: h(x) = (3x + 1) / (x^2 − 2)
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Step 1 — Let f(x) = 3x + 1, so f'(x) = 3
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Step 2 — Let g(x) = x^2 − 2, so g'(x) = 2x
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Step 3 — Apply the quotient rule: [f'(x)g(x) − f(x)g'(x)] / [g(x)]²
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= [3(x^2 − 2) − (3x + 1)(2x)] / (x^2 − 2)^2
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Step 4 — Expand the numerator: (3x^2 − 6) − (6x^2 + 2x) = -3x^2 − 2x − 6
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Result: h'(x) = (-3x^2 − 2x − 6) / (x^2 − 2)^2
7. Chain Rule
Use the chain rule whenever one function is nested inside another — for example, sin(x^2) or (3x + 1)^5. Differentiate the outer function first (leaving the inner function alone), then multiply by the derivative of the inner function: d/dx[f(g(x))] = f'(g(x))·g'(x).
Worked Example: Chain Rule
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Function: h(x) = (3x + 1)^5
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Step 1 — Identify the outer function: ( )^5, and inner function: 3x + 1
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Step 2 — Differentiate the outer function, keeping the inner function unchanged: 5(3x + 1)^4
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Step 3 — Multiply by the derivative of the inner function, d/dx[3x + 1] = 3
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Result: h'(x) = 15(3x + 1)^4
Worked Example: Chain Rule with a Trig Function
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Function: h(x) = sin(x^2)
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Step 1 — Outer function: sin( ), inner function: x^2
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Step 2 — Differentiate the outer function: cos(x^2)
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Step 3 — Multiply by the derivative of the inner function, d/dx[x^2] = 2x
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Result: h'(x) = 2x·cos(x^2)
Trigonometric Derivatives
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | -sin x |
| tan x | sec^2 x |
| cot x | -csc^2 x |
| sec x | sec x · tan x |
| csc x | -csc x · cot x |
Inverse Trigonometric Derivatives
| Function | Derivative |
|---|---|
| arcsin x | 1 / √(1 − x²) |
| arccos x | -1 / √(1 − x²) |
| arctan x | 1 / (1 + x²) |
| arccot x | -1 / (1 + x²) |
| arcsec x | 1 / (|x|·√(x² − 1)) |
| arccsc x | -1 / (|x|·√(x² − 1)) |
Exponential and Logarithmic Derivatives
| Function | Derivative |
|---|---|
| e^x | e^x |
| a^x (a > 0) | a^x · ln a |
| ln x (x > 0) | 1 / x |
| log_a x | 1 / (x · ln a) |
Worked Example: Exponential Chain Rule
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Function: h(x) = e^(3x^2)
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Step 1 — Outer function: e^( ), inner function: 3x^2
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Step 2 — Differentiate the outer function, keeping the inner unchanged: e^(3x^2)
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Step 3 — Multiply by the derivative of the inner function, d/dx[3x^2] = 6x
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Result: h'(x) = 6x·e^(3x^2)
Higher-Order Derivatives
Taking the derivative of a derivative gives the second derivative, written f''(x) or d²y/dx². The second derivative measures how the rate of change itself is changing (concavity and acceleration), and the process can be repeated for third, fourth, and higher-order derivatives. Example: if f(x) = x^4, then f'(x) = 4x^3, f''(x) = 12x^2, and f'''(x) = 24x.
Common Mistakes to Avoid
- Forgetting the chain rule's inner-function multiplier — e.g., writing d/dx[sin(x^2)] as cos(x^2) instead of 2x·cos(x^2).
- Applying the product rule as if derivatives distribute over multiplication (they don't) — always use f'g + fg'.
- Mixing up the quotient rule's numerator order; the sign matters: f'g − fg', not fg' − f'g.
- Treating a^x and x^a as the same differentiation pattern — one needs the exponential rule (with ln a), the other needs the power rule.
- Forgetting that d/dx[ln x] = 1/x has no exponent — it's a common error to write 1/x² or ln(1)/x.
Frequently Asked Questions
Q1What is the difference between a derivative and a rate of change?
A derivative is the precise, instantaneous rate of change of a function at a single point — the slope of the tangent line there — as opposed to an average rate of change measured over an interval.
Q2When should I use the product rule versus the chain rule?
Use the product rule when two separate functions are multiplied together, like x²·sin x. Use the chain rule when one function is nested inside another, like sin(x²). Some problems require both rules combined.
Q3Is the derivative of a constant always zero?
Yes. Since a constant does not change as x changes, its rate of change — and therefore its derivative — is always 0.
Q4How do I find the derivative of a function with a fractional exponent?
Apply the power rule exactly as you would with an integer exponent: bring the exponent down as a coefficient and subtract 1 from it. For example, d/dx[x^(1/2)] = (1/2)x^(-1/2).