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Intermediate Topic 31 of 51 Math for Computing

Derivatives

Rate of change of a function.

How it works

The derivative of a function measures its instantaneous rate of change, defined as the limit of the difference quotient, and geometrically gives the slope of the tangent line. Rules include the power, product, quotient and chain rules, with known derivatives for exponentials, logarithms and trigonometric functions. The sign of the derivative shows where a function increases or decreases, zeros of the derivative find candidate maxima and minima and the second derivative gives concavity.

Where it is used

Optimization of cost functions, gradient descent in machine learning, physics of motion, sensitivity analysis, backpropagation through automatic differentiation and numerical solution of differential equations use derivatives.

Common pitfall

Applying the chain rule incorrectly, forgetting inner derivatives, is the usual slip. Derivatives do not exist at corners or discontinuities. Numerical differentiation by finite differences suffers from rounding and truncation errors, so choose step sizes carefully or use automatic differentiation. A zero derivative does not guarantee an extremum.

Where it fits in the course

Derivatives is topic 31 of 51 in Math for Computing, rated Intermediate. It follows Orthogonality and Projections. The next topic is Partial Derivatives and Gradients.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Math for Computing quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Derivatives?
Rate of change of a function. It belongs to Math for Computing and is an intermediate-level topic.
Where do people use Derivatives?
Optimization of cost functions, gradient descent in machine learning, physics of motion, sensitivity analysis, backpropagation through automatic differentiation and numerical solution of differential equations use derivatives.
What is the most common mistake with Derivatives?
Applying the chain rule incorrectly, forgetting inner derivatives, is the usual slip.
What should I learn before this topic?
Start with Orthogonality and Projections, the topic just before it in Math for Computing.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Math for Computing quiz.
Is Derivatives hard to learn?
It is an Intermediate topic. Be comfortable with the Beginner topics in this area first, then read the common pitfall carefully, since it is where most mistakes happen.
How long does it take to learn Derivatives?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Derivatives?
Orthogonality and Projections, Partial Derivatives and Gradients, Singular Value Decomposition are closely connected. The full list is in the related topics section of this page.
What should I study after Derivatives?
Continue with Partial Derivatives and Gradients, the next topic in Math for Computing.

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