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Μάθημα: AP®︎ Λογισμός BC > Ενότητα 2
Μάθημα 8: Derivatives of cos(x), sin(x), 𝑒ˣ, and ln(x)- Derivatives of sin(x) and cos(x)
- Worked example: Derivatives of sin(x) and cos(x)
- Derivatives of sin(x) and cos(x)
- Proving the derivatives of sin(x) and cos(x)
- Derivative of 𝑒ˣ
- Derivative of ln(x)
- Παράγωγοι του 𝑒ˣ και του ln(x)
- Proof: The derivative of 𝑒ˣ is 𝑒ˣ
- Proof: the derivative of ln(x) is 1/x
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Proving the derivatives of sin(x) and cos(x)
Proving that the derivative of sin(x) is cos(x) and that the derivative of cos(x) is -sin(x).
The trigonometric functions
The AP Calculus course doesn't require knowing the proofs of these derivatives, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.
First, we would like to find two tricky limits that are used in our proof.
1.
2.
Now we are ready to prove that the derivative of
Finally, we can use the fact that the derivative of
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