Differentiation and Derivatives Explained
Differentiation is one of the two central operations of calculus (the other being integration). It provides a precise mathematical tool for finding the rate at which one quantity changes with respect to another — the gradient of a curve at any chosen point. Its applications span physics, economics, engineering, biology, and computer science.
The Gradient of a Curve
For a straight line, the gradient is constant and equals rise over run. For a curve, the gradient changes from point to point. The gradient at any particular point on a curve is defined as the gradient of the tangent to the curve at that point.
To find this gradient algebraically, consider two points on the curve y = f(x): the point (x, f(x)) and a nearby point (x + h, f(x + h)). The gradient of the chord joining them is:
[f(x + h) − f(x)] / h
As h approaches zero, the chord becomes the tangent, and the gradient becomes the derivative:
f'(x) = limh→0 [f(x + h) − f(x)] / h
This is differentiation from first principles. The derivative is written as f'(x), dy/dx, or d/dx[f(x)], depending on context.
The Power Rule
For any power of x, the derivative follows a simple pattern: bring the exponent down as a coefficient and reduce the exponent by one.
d/dx [xn] = nxn−1
This works for all real values of n, including negative and fractional powers:
- d/dx [x3] = 3x2
- d/dx [x5] = 5x4
- d/dx [x−2] = −2x−3
- d/dx [x1/2] = (1/2)x−1/2
- d/dx [c] = 0 (the derivative of a constant is zero)
The derivative of a sum is the sum of the derivatives, and constants multiply through: d/dx [5x3 + 2x − 7] = 15x2 + 2.
The Chain, Product, and Quotient Rules
Three rules extend differentiation to more complex expressions.
Chain rule — for a function of a function, y = f(g(x)):
dy/dx = f'(g(x)) × g'(x) (differentiate the outer function, then multiply by the derivative of the inner function)
Example: y = (3x + 1)5. Let u = 3x + 1, so y = u5. Then dy/dx = 5u4 × 3 = 15(3x + 1)4.
Product rule — for y = u(x) × v(x):
dy/dx = u'v + uv'
Example: y = x2 × ex. Then dy/dx = 2x × ex + x2 × ex = ex(2x + x2).
Quotient rule — for y = u(x) / v(x):
dy/dx = (u'v − uv') / v2
Mnemonic: "low d-high minus high d-low, square the bottom and away we go."
Standard Derivatives to Know
| f(x) | f'(x) |
|---|---|
| xn | nxn−1 |
| ex | ex |
| ekx | kekx |
| ln(x) | 1/x |
| sin(x) | cos(x) |
| cos(x) | −sin(x) |
| tan(x) | sec2(x) |
Finding Turning Points
A turning point (stationary point) is where the gradient is zero: dy/dx = 0. To find and classify turning points:
- Differentiate the function to get dy/dx.
- Set dy/dx = 0 and solve for x to find the x-coordinate(s) of the turning point(s).
- Substitute back into the original function to find the corresponding y-coordinate(s).
- Find the second derivative d2y/dx2. If it is positive at the turning point, the point is a minimum; if negative, it is a maximum; if zero, further investigation is needed.
Example: y = x3 − 3x. Then dy/dx = 3x2 − 3 = 3(x2 − 1). Setting this to zero gives x = ±1. At x = 1: d2y/dx2 = 6x = 6 > 0, so a minimum. At x = −1: d2y/dx2 = −6 < 0, so a maximum.
Real-World Applications
Differentiation is not just algebra — it describes change in the physical world. Velocity is the derivative of displacement with respect to time (v = ds/dt); acceleration is the derivative of velocity (a = dv/dt). In economics, marginal cost is the derivative of total cost with respect to quantity produced. In biology, population growth rate is modelled using derivatives. In machine learning, gradient descent algorithms use derivatives to minimise a loss function by moving in the direction of steepest descent — the same idea as sliding down to the bottom of a bowl.
Summary
The derivative f'(x) measures the instantaneous rate of change of f at x, defined as the limit of the gradient of a chord as its length approaches zero. The power rule handles polynomial terms; the chain, product, and quotient rules extend differentiation to composite, multiplied, and divided functions. Setting dy/dx = 0 locates turning points, and the sign of the second derivative classifies them as maxima or minima. These techniques underpin vast areas of applied mathematics, from mechanics and thermodynamics to finance and artificial intelligence.