Differentiating Polynomials and Tangent Lines

The derivative by rule

The derivative of a function tells us the slope of its graph at every point. At Ordinary Level you don't work it out from a limit. You use a rule.

We write the derivative as $f'(x)$ or as $\dfrac{dy}{dx}$ when $y = f(x)$. Differentiating $f'(x)$ again gives the second derivative, written $f''(x)$ or $\dfrac{d^2y}{dx^2}$.

Why does the rule make sense? A constant never changes, so its slope is $0$. The line $y = 5x$ has slope $5$ everywhere, and the rule gives $5 \cdot 1 \cdot x^0 = 5$. The curve $y = x^2$ gets steeper as $x$ grows, and its slope $2x$ also grows.

To find the derivative's value at a point, put the $x$-value into $f'(x)$. For example, if $f'(x) = 3x^2 - 6$, then $f'(2) = 3(4) - 6 = 6$.

The slope of a curve at a point

On a straight line the slope is the same everywhere. On a curve it changes from point to point. The slope of the curve at a point is the slope of the tangent there, which is the straight line that just touches the curve at that point.

Drag the slider below. The curve is $y = x^2$ and the straight line is its tangent at $x = a$. Compare the slope of the tangent with $2a$, which is $f'(a)$.

We can also work backwards. If we are told the slope, we set the derivative equal to that number and solve for $x$.

The equation of a tangent

A tangent is a straight line, so we can use the formula for a line through a point with a given slope: $y - y_1 = m(x - x_1)$. We need two things:

1. a point $(x_1, y_1)$ on the curve, found from $f(x_1)$;

2. the slope $m = f'(x_1)$.

A tangent can meet the curve again somewhere else. To find where, set the line equal to the curve and solve. The point where the line touches the curve is always a repeated root, so you can use it to check your factorising.

Sometimes the unknown is in the equation of the curve. A point on the curve gives one equation, because the point must fit the function. The slope of the tangent gives another, because $f'(x)$ must equal that slope.

Rates of change and turning points

The derivative tells us how fast one quantity changes with respect to another. If $h(t)$ is the height of an object at time $t$, then $h'(t)$ is its rate of change of height, which is its speed. A negative value means the height is falling.

At the top of a curve, or the bottom of a dip, the tangent is horizontal, so the slope is $0$. These points are called turning points (or stationary points).

To find and classify turning points:

Maximum and minimum problems

Many real problems ask for the largest area or the smallest cost. The steps are always the same.

Now we can answer the opening question. If the sliotar's height is $h(t)$, its speed is $h'(t)$. At the top the speed is $0$, so we solve $h'(t) = 0$ and then put that time into $h(t)$ to get the greatest height.

Summary

• Differentiate by rule: $ax^n \to anx^{n-1}$. Constants become $0$, and sums are done term by term.

• $f'(a)$ is the slope of the curve (and of its tangent) at $x = a$. Use $f'(x)$ for slopes and $f(x)$ for $y$-values.

• Tangent at $x = a$: find the point $(a, f(a))$ and the slope $f'(a)$, then use $y - y_1 = m(x - x_1)$.

• If a tangent is given, the point of contact fits both the curve and the line, and the slope of the curve there equals the slope of the line.

• Turning points: solve $f'(x) = 0$. If $f''(x) > 0$ it is a local minimum, and if $f''(x) < 0$ it is a local maximum.

• Rates of change: $h'(t)$ is the speed. For a maximum or minimum problem, write the quantity in one variable, differentiate, set it to $0$, check with $f''$, and answer in context.