Derivatives and rates of change

Review slopes, increments and units

For two points on a graph, slope is vertical change divided by horizontal change. If the input changes from a to a+h, the input increment is h and the output increment is f(a+h)−f(a). Use parentheses: subtracting f(a) means subtracting the entire expression. The average rate is [f(a+h)−f(a)]/h and is defined only for h≠0, with both inputs in the domain.

If the input is time in seconds and the output is position in metres, the rate has unit m/s. If the input is the number of products and the output is cost in monetary units, the rate is cost per product. Thus, a derivative is not just a number associated with a graph: it expresses one quantity's sensitivity to another. Changing the horizontal unit changes that rate's numerical value.

From secant to tangent

A secant passes through two distinct points of the curve. Keep one point fixed, at a, and bring the other closer by letting h tend to zero. If the slopes converge to the same finite number from the permitted sides, that number is f′(a), the derivative at a. The usual definition at an interior point requires a two-sided limit.

We do not substitute h=0 into the quotient: that would generally give 0/0. We simplify for h≠0 and only then take the limit. The tangent is the line y=f(a)+f′(a)(x−a), preserving the point and local slope. It may cross the curve or meet it again elsewhere. The idea of touching only once, useful for some circles, is not the general definition of a tangent.

Deduce a rule before using it

For f(x)=x², calculate f(a+h)=(a+h)²=a²+2ah+h². Subtracting a² and dividing by h, for h≠0, gives 2a+h. The limit is 2a. Since a can be any real number, the derivative function is f′(x)=2x. For a constant, the numerator is zero and the derivative is zero. For f(x)=mx+b, the numerator is mh and the derivative is m.

The power rule generalises the result: the derivative of xⁿ is nxⁿ⁻¹, at points where the function and real differentiation make sense. For positive integer exponents, expanding (x+h)ⁿ leaves, after division by h, a term nxⁿ⁻¹ and terms with factors h that vanish in the limit. The rule also works for many noninteger exponents, but the domain must be analysed: √x has derivative 1/(2√x) only for x>0.

Combine rates: sum, product and composition

The derivative of a sum is the sum of the derivatives; a constant multiplier stays with the derivative. In a product uv, the change includes the change in u with v nearly fixed and the change in v with u nearly fixed. This leads to (uv)′=u′v+uv′. It is not u′v′. For a quotient, (u/v)′=(u′v−uv′)/v², where v≠0 and the functions are differentiable.

A composition describes one transformation inside another. If y=g(u) and u=f(x), a small change in x produces approximately f′(x) times that change in u; then g′(u) multiplies it. Thus, [g(f(x))]′=g′(f(x))f′(x). For (3x+1)², the outer factor is 2(3x+1) and the inner factor is 3. The result 6(3x+1) can be confirmed by expanding the polynomial. These two approaches provide a useful check.

When the derivative fails

The function |x| is continuous at zero, but the one-sided slopes are −1 and 1. There is no unique derivative at that point. Continuity does not guarantee differentiability. The converse implication is valid: if a finite derivative exists at a, then f is continuous at a. Indeed, the change f(a+h)−f(a) is the difference quotient multiplied by h; the first factor tends to a finite number and the second to zero.

A discontinuity prevents differentiability at the point. A vertical tangent also does not correspond to a finite real derivative, although an unbounded slope can be studied. At a domain endpoint there may be a one-sided derivative; it must be identified as such. When differentiating 1/x, the formula −1/x² does not allow x=0: the original function does not even exist at that input. The derivative's domain and the function's domain may differ.

Read rates in context and data

If s(t) represents position, s′(t) is velocity, with its sign depending on the axis; speed is |s′(t)|. Acceleration is the derivative of velocity, not velocity itself. A negative derivative means the output is decreasing, not necessarily that the output is negative. A tank can contain a lot of liquid while its volume decreases.

With measured data, an instantaneous rate is estimated from rates over small intervals. Reducing the interval too much can amplify measurement errors: dividing a small noisy difference by a tiny interval makes the estimate unstable. For a smooth function, the central difference [f(a+h)−f(a−h)]/(2h) can be useful, but remains an approximation. Finally, a zero derivative at one instant does not mean the function is constant: t² has derivative zero at zero and varies in every neighbourhood. It is the behaviour over an interval that allows us to conclude constancy.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

The derivative of a product is the product of the derivatives.

Each factor contributes to the change; use u′v+uv′.

A zero derivative implies a maximum or minimum.

It is only a candidate; x³ has derivative zero at 0 and remains increasing.

A continuous function always has a derivative.

A corner such as |x| at zero prevents a unique slope from existing.