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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB > Unit 2

Lesson 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)
- Derivatives of 𝑒ˣ and ln(x)
- Proof: The derivative of 𝑒ˣ is 𝑒ˣ
- Proof: the derivative of ln(x) is 1/x

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# Derivatives of sin(x) and cos(x)

Now we explore the intuition behind the derivatives of trigonometric functions, discovering that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x). By analyzing tangent line slopes, we gain a deeper understanding of these fundamental relationships.

## Want to join the conversation?

- Regarding the unit circle, when sin is 0 cos is 1 or -1. But sin's slope will be 0/ infinity or 0/ - infinity. Both undefined and not cos. Not sure what im doing wrong here.(4 votes)
- Remember, a derivative involves instantaneous slope, but that doesn't mean that you're dividing by zero - if so, then we couldn't take the derivative of any function! You should try reviewing the videos on derivatives to get a feel for the intuition of what instantaneous slope really means.(17 votes)

- At which points are the tangents of cos x and sin x parallel ? What makes them important ?(5 votes)
- Their tangent lines are parallel when their derivatives are equal, that is, -sin(x) = cos(x), which occurs at 3pi / 4 and 7pi / 4.

Important? I'm honestly not sure what the application might be, except for that it's useful in general for trigonometry.(4 votes)

## Video transcript

- [Instructor] What I'd
like to do in this video is get an intuitive sense
for what the derivative with respect to x of sine of x is and what the derivative with
respect to x of cosine of x is. And I've graphed y is equal
to cosine of x in blue and y is equal to sine of x in red. We're not going to prove
what the derivatives are, but we're gonna know what they
are, get an intuitive sense and in future videos
we'll actually do a proof. So let's start with sine of x. So the derivative can be viewed as the slope of the tangent line. So for example at this
point right over here, it looks like the slope of our
tangent line should be zero. So our derivative function
should be zero at that x value. Similarly, over here, it looks
like the derivative is zero. Slope of the tangent line would be zero. So whatever our derivative
function is at that x value, it should be equal to zero. If we look right over here on sine of x, it looks like the slope
of the tangent line would be pretty close to one. If that is the case, then
in our derivative function when x is equal to zero that derivative function
should be equal to one. Similarly, over here, it looks like the slope of the tangent
line is negative one, which tells us that
the derivative function should be hitting the value of
negative one at that x value. So you're probably seeing
something interesting emerge. Everywhere, while we're trying to plot the slope of the tangent
line, it seems to coincide with y is equal to cosine of x. And it is indeed the case that
the derivative of sine of x is equal to cosine of x. And you can see that it makes sense, not just at the points we
tried, but even in the trends. If you look at sine of x
here, the slope is one, but then it becomes less
and less and less positive all the way until it becomes zero. Cosine of x, the value
of the function is one and it becomes less and less positive all the way until it equals zero. And you could keep doing
that type of analysis to feel good about it. In another video we're going
to prove this more rigorously. So now let's think about cosine of x. So cosine of x, right over here, the slope of the tangent line looks like it is zero. And so it's derivative function needs to be zero at that point. So, hey, maybe it's sine of x. Let's keep trying this. So over here, cosine of x, it looks like the slope of the
tangent line is negative one and so we would want the
derivative to go through that point right over there. All right this is starting to seem, it doesn't seem like the
derivative of cosine of x could be sine of x. In fact, this is the opposite
of what sine of x is doing. Sine of x is at one, not
negative one at that point. But that's an interesting theory, maybe the derivative of cosine
of x is negative sine of x. So let's plot that. So this does seem to coincide. The derivative of cosine of x
here looks like negative one, the slope of a tangent line and negative sign of this
x value is negative one. Over here the derivative of cosine of x looks like it is zero and negative sine of x is indeed zero. So it actually turns
out that it is the case, that the derivative of cosine
of x is negative sine of x. So these are really good to know. These are kind of fundamental trigonometric derivatives to know. We'll be able to derive
other things for them. And hopefully this video gives
you a good intuitive sense of why this is true. And in future videos, we
will prove it rigorously.