Derivatives and Rate of Change

A straight line has one slope everywhere. A curve does not — its steepness changes from point to point. The derivative is what gives you the slope at a single point on a curve, and that same number is the instantaneous rate of change of the function there.

Concept What a derivative is

The derivative is the limit of the average rate of change as the interval shrinks to zero.

Geometrically — the derivative is the slope of the tangent line touching the curve at that point.

Practically — it is how fast the function's value is changing with respect to the input.

Concept The basic rules

A handful of rules let you differentiate without going back to the limit definition each time.

Power rule —

\frac{d}{dx}\,x^n = n\,x^{\,n-1}

. Bring the exponent down in front, then reduce it by one.

Constant rule — the derivative of any constant is 0, since a constant never changes.

Sum rule — differentiate a sum term by term.

Constant multiple rule — a number multiplying a term simply carries through.

Example Differentiating a quadratic

Find the derivative of $f(x) = x^2 + 3x - 2$ .

Apply the power rule to

x^2

: the exponent 2 comes down, leaving

2x^{1} = 2x

For

3x

, the exponent is 1, so it becomes

3

The constant

-2

differentiates to 0.

Add the results term by term.

⟹ f ′(x) = 2x + 3

The derivative is itself a function. Feed it a value of $x$ and it returns the slope of the curve at that point.

Example Using the derivative at a point

For that same function, find the slope of the curve at $x = 1$ and at $x = -2$ .

f'(1) = 2(1) + 3 = 5

— the curve is rising steeply there.

f'(-2) = 2(-2) + 3 = -1

— a negative slope, so the curve is falling.

⟹ slopes of 5 and −1

This is the essential difference from a straight line. The same curve is climbing at one point and descending at another, and the derivative reports which.

Note Reading the sign

A positive derivative means the function is increasing at that point.

A negative derivative means it is decreasing.

A derivative of zero means the tangent is horizontal — often a peak or a trough.

The size matters as well as the sign: a larger absolute value means a steeper curve at that point.

Note Mistakes to avoid

Forgetting to reduce the exponent after bringing it down.

Leaving a constant in the answer instead of differentiating it to 0.

Treating the derivative as a single number when it is a function of

x

Expecting a curve to have one slope everywhere, as a straight line does.

Dropping a coefficient that should carry through unchanged.

Summary

The derivative gives the slope of a curve at a single point, and equals the instantaneous rate of change.
Geometrically it is the slope of the tangent line touching the curve there.
The power rule brings the exponent down and reduces it by one; constants differentiate to 0.
Sums are differentiated term by term, and constant multipliers carry through.
A positive derivative means increasing, negative means decreasing, and zero marks a horizontal tangent.