The gradient of a chord
Differentiation is all about gradient. A chord is a straight line joining two points on a curve. The closer the points get, the closer the chord gradient gets to the true gradient of the curve at that point.
As you zoom in, notice the chord and the curve almost lie on top of one another - but they are never exactly the same. The chord is straight; the curve bends. Differentiation finds what the chord gradient tends towards as the gap becomes infinitesimally small.
Generalising
Instead of two specific numbers, use two general points. Let the first be at and the second a small distance further along, at . Their -values are then and .
- The gap between the two points:
- Gradient of the chord is change in over change in :
- Shrink the gap. Make smaller and smaller. The chord rotates towards the tangent - but is never actually zero.
- Take the limit. The value the chord gradient tends towards as is called the derivative:
This is the idea behind differentiation from first principles. The full proof with worked examples is on the First Principles tab.
The rule for differentiating
Multiply by 1, drop the power to 0. Since , the simply disappears - the gradient of is always 1.
A constant is . Multiplying by the power 0 wipes it out - every constant differentiates to 0 (a flat line has zero gradient).
Sums: differentiate term by term
Apply the rule to each term separately - the sum and difference rule means you never have to deal with a whole expression at once.
Differentiation notation - , and
Lagrange notation - . The derivative of is written ("f-dash x"). The second derivative is - covered on the Stationary Points tab.
Leibniz notation - . If the function is written , its derivative is ("dee-y by dee-x") - the rate of change of with respect to .
Operator notation - . On its own, is an instruction: "differentiate whatever follows with respect to ." So .
Worked examples
The key step is almost always to rewrite as a power of before applying the rule.
Questions
Sketching a gradient function
You can sketch the shape of from the graph of without any algebra.
A point where the gradient is zero, - the curve is momentarily flat. On the gradient function these appear as zeros (the graph crosses or touches the -axis).
- Mark the stationary points. Find every place the curve is flat (). Directly below each one, the gradient function sits on the -axis.
- Positive or negative gradient? Uphill (left to right) means - highlighted in green. Downhill means - highlighted in red.
- Is the gradient increasing or decreasing? Where the curve steepens, is rising; where it flattens, is falling. This fixes the shape between the axis crossings.
Increasing & decreasing functions
A function is increasing where and decreasing where . Select a function, toggle the highlights, then read the working.
Method
Tangents & normals
Drag the point along the curve. The gradient is found by differentiating.
Tangent & Normal Definitions
Tangent: the straight line that just touches the curve at a point and has the same gradient as the curve there. Its gradient is .
Normal: the straight line at the same point that is perpendicular to the tangent. Its gradient is the negative reciprocal.
Stationary points
A stationary point has . To classify it we look at the second derivative - the gradient of the gradient.
Types of stationary point
Practice
Points of inflection
Concave & Convex
Concavity describes how the gradient itself is changing as you move left to right. The curve is concave when the gradient is decreasing (each tangent is less steep than the last, so ), and convex when the gradient is increasing (so ). A point of inflection is where the gradient stops increasing and starts decreasing (or vice versa) - so the concavity flips and changes sign.
The curve bends downward. The gradient is decreasing, so .
The curve bends upward. The gradient is increasing, so .
A point of inflection is where the curve changes from concave to convex (or vice versa).
Non-stationary points of inflection
An inflection is where and changes sign. That can happen mid-slope where .
Third derivative method: if and at a point, that point is a non-stationary point of inflection.
Practice
The gradient function
The derivative is itself a function - it gives the gradient of the curve at every value of .
What the two graphs are telling you
Differentiation from first principles
Every differentiation rule comes from this definition. We take two points on , find the gradient of the chord, and ask what value that gradient tends towards as the gap closes - the limit as .
Type a value or tap −/+. Make smaller and watch the chord tend towards the tangent.
You can also drag the red point to change h.