What Is a Derivative?
A derivative measures how fast something is changing at a single instant, rather than on average over an interval. Graphically it is the slope of the tangent line at one point on a curve.
From average to instantaneous
The slope between two points on a curve is an average rate of change: rise over run. Slide the second point closer and closer to the first and that average approaches a single number. That limit is the derivative, and it describes the curve at exactly one point.
Notation
- f'(x) is read f prime of x: the derivative of the function f.
- dy/dx is the derivative of y with respect to x. It is a single symbol, not a fraction.
- Both mean the same thing; different courses prefer different notation.
The power rule
For f(x) = x^n, the derivative is f'(x) = n x^(n-1). Bring the exponent down as a multiplier and reduce the exponent by one. Almost every derivative you meet early on is this rule plus a constant multiplier.
Worked examples
Example 1
Question
Find the derivative of f(x) = x^2 and evaluate it at x = 3.
Answer
f'(x) = 2x, so f'(3) = 6
Steps
- 1Apply the power rule with n = 2: bring the 2 down and reduce the exponent to 1.
- 2f'(x) = 2x^1 = 2x.
- 3Substitute x = 3: f'(3) = 6.
- 4Meaning: at the point (3, 9) the curve is climbing 6 units of height per unit across.
The derivative is a function, not a number. It gives a different slope at every x, and you get a number only by substituting one.
Example 2
Question
Find the derivative of f(x) = 3x^4.
Answer
f'(x) = 12x^3
Steps
- 1The constant multiplier 3 stays and multiplies the result.
- 2Apply the power rule to x^4: it becomes 4x^3.
- 3Multiply by the 3 in front: 3 x 4x^3 = 12x^3.
Example 3
Question
An object's position is s(t) = 5t^2 metres after t seconds. Find its velocity at t = 2.
Answer
20 m/s
Steps
- 1Velocity is the derivative of position with respect to time.
- 2Differentiate: s'(t) = 5 x 2t = 10t.
- 3Substitute t = 2: s'(2) = 20.
- 4The units come from the original: metres per second.
This is why derivatives matter in physics. Position differentiates to velocity, and velocity differentiates again to acceleration.
Common mistakes
Treating the derivative as a number rather than a function.
f'(x) is a function. f'(3) is a number. Questions ask for one or the other, so read carefully.
Differentiating a constant to get the constant.
The derivative of any constant is zero, because a constant never changes.
Reducing the exponent but forgetting to bring it down as a multiplier.
Both happen. x^5 becomes 5x^4, not x^4.
Using the power rule on something like e^(2x).
The power rule applies to a variable raised to a constant power. Exponentials and composites need their own rules.
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. Differentiate f(x) = x^5.
Show answer
f'(x) = 5x^4.
2. Differentiate f(x) = 7x.
Show answer
f'(x) = 7. A straight line has a constant slope.
3. Differentiate f(x) = x^2 + 3x and evaluate at x = 1.
Show answer
f'(x) = 2x + 3, so f'(1) = 5.
Questions people ask
- What does the derivative mean in plain words?
- How fast the output changes when the input changes, right at that moment. Speed is the derivative of distance, and it is the clearest everyday example.
- Why do we need limits to define it?
- Because slope needs two points, and we want the slope at one. The limit is the machinery that makes a one-point slope meaningful.
- What does a negative derivative mean?
- The function is decreasing at that point. The steeper the negative value, the faster it is falling.
- What if the derivative is zero?
- The curve is momentarily flat: a maximum, a minimum, or a point of inflection. Finding those points is what optimisation problems are built on.
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