# Standard derivatives

Know these by heart: sin x gives cos x, cos x gives -sin x, e^x gives e^x, and ln x gives 1/x for x > 0.

Maths (university) · Part 03: Differentiation · Card 8 of 10 · Last updated 2026-10-19

## Analogy

Like times tables: memorised once, used constantly.

## Example

**d/dx of 5e^x - 2cos x**

5e^x stays 5e^x.
-2cos x becomes -2(-sin x).
Result: 5e^x + 2 sin x.
At x = 0: 5 + 0 = 5.

## In practice

They are the building blocks that the rules combine for any complicated function.

## Practice

### Flashcards

- **What is d/dx of cos x?** -sin x.
- **What is d/dx of ln x?** 1/x, for x > 0.

## Questions people ask

### Why is e to the x its own derivative?

The number e is defined so that e to the x grows at a rate equal to its current size. Its slope at any point equals its height. At x = 0 both are 1. No other base gives exactly this property, which makes e special.

### What is the derivative of the natural logarithm?

The derivative of ln x is 1 over x for x greater than 0. This fits ln growing quickly at first and slowly later. At x = 1 the slope is 1, while at x = 10 it has dropped to 0.1.

## Related concepts

- [Derivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-1)
- [Tangent line](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-2)
- [Rate of change](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-3)
- [Power rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-4)

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