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Proving the derivatives of sin(x) and cos(x)

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.A (LO)
,
FUN‑3.A.4 (EK)
​Proving that the derivative of sin(x) is cos(x) and that the derivative of cos(x) is -sin(x).
The trigonometric functions sine, left parenthesis, x, right parenthesis and cosine, left parenthesis, x, right parenthesis play a significant role in calculus. These are their derivatives:
ddx[sin(x)]=cos(x)ddx[cos(x)]=sin(x)\begin{aligned} \dfrac{d}{dx}[\sin(x)]&=\cos(x) \\\\ \dfrac{d}{dx}[\cos(x)]&=-\sin(x) \end{aligned}
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. limit, start subscript, x, \to, 0, end subscript, start fraction, sine, left parenthesis, x, right parenthesis, divided by, x, end fraction, equals, 1

Khan Academy video wrapper
Limit of sin(x)/x as x approaches 0See video transcript

2. limit, start subscript, x, \to, 0, end subscript, start fraction, 1, minus, cosine, left parenthesis, x, right parenthesis, divided by, x, end fraction, equals, 0

Khan Academy video wrapper
Limit of (1-cos(x))/x as x approaches 0See video transcript

Now we are ready to prove that the derivative of sine, left parenthesis, x, right parenthesis is cosine, left parenthesis, x, right parenthesis.

Khan Academy video wrapper
Proof of the derivative of sin(x)See video transcript

Finally, we can use the fact that the derivative of sine, left parenthesis, x, right parenthesis is cosine, left parenthesis, x, right parenthesis to show that the derivative of cosine, left parenthesis, x, right parenthesis is minus, sine, left parenthesis, x, right parenthesis.

Khan Academy video wrapper
Proof of the derivative of cos(x)See video transcript

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