Derivative vs Integral
They are opposite operations on the same relationship. A derivative takes a total and tells you the rate; an integral takes a rate and tells you the total. Almost every mix-up between them is really a question about which one you were given.
The short version
- Derivative: rate of change at an instant. Input a function, output its slope function.
- Integral: accumulated total over an interval. Input a rate, output the amount built up.
- Differentiating and then integrating returns you to where you started, up to a constant.
The physical picture
Position differentiates to velocity, and velocity differentiates to acceleration. Go the other way and acceleration integrates to velocity, and velocity integrates to position. Same three quantities, read in opposite directions.
Which does the question want?
- How fast, at what rate, slope, or marginal: derivative.
- Total, area under, accumulated, or net change: integral.
- Maximum or minimum: derivative, because those are where the slope is zero.
- Given a rate, find the amount: integral.
Worked examples
Example 1
Question
Differentiate x^3, then integrate the result. What comes back?
Answer
x^3 + C, the original function up to a constant
Steps
- 1Differentiate x^3 using the power rule: 3x^2.
- 2Now integrate 3x^2: raise the exponent and divide, giving 3x^3 / 3 = x^3.
- 3Add the constant: x^3 + C.
- 4The round trip returns the original function, except that the constant cannot be recovered.
This is the fundamental theorem of calculus in miniature: the two operations undo each other.
Example 2
Question
f(x) = x^2. Compare f'(2) with the definite integral of f'(x) from 1 to 3.
Answer
f'(2) = 4, and the integral is 8
Steps
- 1The derivative is f'(x) = 2x, so f'(2) = 4. That is the slope at a single point.
- 2The definite integral of 2x from 1 to 3 uses the antiderivative x^2.
- 3Evaluate: 3^2 - 1^2 = 9 - 1 = 8.
- 4That 8 is the net change in f between x = 1 and x = 3, not a slope.
- 5So one is an instantaneous rate and the other is a total change. Different questions, different numbers.
Example 3
Question
Water flows into a tank at 6t litres per minute. How much enters in the first 4 minutes, and how fast is the rate rising?
Answer
48 litres in total, with the rate rising by 6 litres per minute each minute
Steps
- 1The total accumulated is an integral: the antiderivative of 6t is 3t^2.
- 2Evaluate from 0 to 4: 3(16) - 0 = 48 litres.
- 3How fast the rate itself is changing is a derivative: the derivative of 6t is 6.
- 4So the rate increases by 6 litres per minute for every minute that passes.
- 5Same function, two different questions, two different operations.
Common mistakes
Integrating when the question asks how fast something is changing.
Rate questions are derivative questions. Look for how fast, per unit, or slope in the wording.
Differentiating a rate to find a total.
Totals accumulate, so they integrate. Differentiating a rate gives you how the rate itself is changing.
Thinking the two operations are unrelated topics.
They are inverses. Checking an integral by differentiating uses exactly that relationship.
Expecting the round trip to return the original exactly.
It returns it up to a constant, because differentiating erased any constant that was there.
Practice it
Work these out first, then open the answer. Stuck halfway? Select the problem and ask Solvecoon for the step you are missing.
1. You are given a velocity function and asked for distance travelled. Which operation?
Show answer
Integrate. Distance accumulates velocity over time.
2. You are given a position function and asked for speed at t = 5. Which operation?
Show answer
Differentiate, then substitute t = 5.
3. Differentiate 4x^3, then integrate your answer.
Show answer
The derivative is 12x^2, and integrating that gives 4x^3 + C.
Questions people ask
- Which should I learn first?
- Derivatives, because integration is usually taught as reversing them, and every integral can be checked by differentiating.
- Is an integral just an antiderivative?
- An indefinite integral is. A definite integral is a number, computed from an antiderivative by evaluating at both bounds.
- What is the fundamental theorem of calculus?
- The statement that these two operations are inverses, which is what lets you evaluate an area by finding an antiderivative instead of summing rectangles.
- Can I differentiate and integrate the same function?
- Yes, and doing both is a good habit. Differentiating your integral is the fastest way to check it.
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