Part 03 · Differentiation · Card 8 of 10
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.
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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.