Derivatives
Start gently
Differentiation is the tool for finding the slope at a single instant. It is the steepness of the speed graph you drew in primary school, and the rate of change of a linear function from junior high, sharpened down to one point on a curve.
Key points
The definition of a derivative
f'(x) is the slope of the function at the single point x — its steepness at that instant. You get there by taking the average slope over a small step Δx and then letting that step shrink toward zero.
Differentiating a power of x
This is the formula you will use more than any other. To differentiate x to the power n, bring the exponent down in front as a multiplier and then drop the exponent by one.
What the derivative tells you (slope, maximums, and minimums)
The sign of the derivative reads the shape of the graph: f'(x) > 0 means the function is climbing, f'(x) < 0 means it is falling, and f'(x) = 0 marks a spot where it has levelled off — a candidate for a peak or a valley. Setting the derivative of a profit function to zero and solving is exactly how "maximize profit" is done in mathematics.
The product rule and the quotient rule
These two rules handle a product or a quotient of functions, which comes up constantly in practice — differentiating a revenue function built as price times quantity, for example.
Differentiating a function inside a function (the chain rule)
Split the expression into an outer function and an inner one, differentiate each, and multiply the results. Backpropagation — the algorithm that trains every neural network — is nothing more than this rule applied layer after layer.
See it drawn
Since y' = 2x, the slope at x = 2 is 4. The tangent is the line through that point with that slope.
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