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

\sin (x)

and

\cos (x)

play a significant role in calculus. These are their derivatives:

\begin{aligned} \frac{d}{d x} [\sin (x)] & = \cos (x) \\ \frac{d}{d x} [\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. $lim_{x \to 0} \frac{\sin (x)}{x} = 1$ ‍

Khan Academy video wrapper

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

2. $lim_{x \to 0} \frac{1 - \cos (x)}{x} = 0$ ‍

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 $\sin (x)$ ‍ is $\cos (x)$ ‍.

Khan Academy video wrapper

Proof of the derivative of sin(x)See video transcript

Finally, we can use the fact that the derivative of $\sin (x)$ ‍ is $\cos (x)$ ‍ to show that the derivative of $\cos (x)$ ‍ is $- \sin (x)$ ‍.

Khan Academy video wrapper

Proof of the derivative of cos(x)See video transcript

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Хан Академийн англи хуудас дээр ямар хэлэлцүүлэг өрнөж байгааг үзэхийг хүсвэл энд дарна уу.

AP®︎ Calculus AB

Course: AP®︎ Calculus AB > Unit 2

First, we would like to find two tricky limits that are used in our proof.

1. limx→0sin⁡(x)x=1‍

2. limx→01−cos⁡(x)x=0‍

Now we are ready to prove that the derivative of sin⁡(x)‍ is cos⁡(x)‍ .

Finally, we can use the fact that the derivative of sin⁡(x)‍ is cos⁡(x)‍ to show that the derivative of cos⁡(x)‍ is −sin⁡(x)‍ .

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1. $lim_{x \to 0} \frac{\sin (x)}{x} = 1$ ‍

2. $lim_{x \to 0} \frac{1 - \cos (x)}{x} = 0$ ‍

Now we are ready to prove that the derivative of $\sin (x)$ ‍ is $\cos (x)$ ‍.

Finally, we can use the fact that the derivative of $\sin (x)$ ‍ is $\cos (x)$ ‍ to show that the derivative of $\cos (x)$ ‍ is $- \sin (x)$ ‍.