# How Do You Differentiate sin, cos, and tan Functions?

• EIRE2003
In summary, the rules for differentiating tan, sin, and cos are as follows: cos = -sin, tan = sin/cos, and you can derive sin's derivative using the limit definition or the quotient rule. The derivatives for the inverse trig functions can also be found using these rules.
EIRE2003
What are the rules for differentiating tan, sin & cos?

I know cos = -sin

tan = sin/cos?

Be careful of what you type.

You can derive sin's derivative (say that five times fast) using the limit definition, from there a simple Taylor series expansion will get you the other two, or knowing sinx' = cosx and cosx' = -sinx, you can use the quotient rule to find tanx.

Yea, just like what Whozum said...

if $$f(x) = \sin \theta$$,$$f'(x) = \cos \theta$$
$$f(x) = \cos \theta$$, $$f'(x) = -\sin \theta$$
$$f(x) = \tan \theta$$, $$f'(x) = \sec^2 \theta$$
$$f(x) = \csc \theta$$, $$f'(x) = -\csc\theta \cot\theta$$
$$f(x) = \sec \theta$$, $$f'(x) = \sec\theta \tan\theta$$
$$f(x)= \cot\theta$$, $$f'(x) = -\csc^2\theta$$

Try finding the derivative of the inverse trig functions

You are correct $$\frac{d}{dx}\cos{x} = -\sin{x}$$

,and I assume you know that $$\frac{d}{dx}\sin{x} = \cos{x}$$.

Using these two definitions, use the quotient rule to find the derivative of tanx as Whozum said above.

$$\frac{d}{dx}\frac{u}{v} = \frac{vu' - uv'}{v^2}$$

Express tangent as $$\frac{\sin{x}}{\cos{x}}$$ and see what you get using the rule above.

Jameson

## 1. What is differentiation?

Differentiation is a mathematical process used to find the rate of change of a function at a specific point. It involves finding the derivative of a function, which represents the slope of the tangent line to the function at that point.

## 2. Why is differentiation important?

Differentiation is important because it allows us to analyze how a function is changing at a specific point. This is useful in many fields, such as physics, economics, and engineering, where understanding rates of change is crucial.

## 3. How do you differentiate a function?

To differentiate a function, you need to use the rules of differentiation, which include the power rule, product rule, quotient rule, and chain rule. These rules allow you to find the derivative of any function, as long as the function is differentiable.

## 4. What is the difference between differentiation and integration?

While differentiation involves finding the rate of change of a function, integration involves finding the cumulative effect of a function over a given interval. In other words, differentiation is like zooming in on a function, while integration is like zooming out.

## 5. Can differentiation be applied to all functions?

No, differentiation can only be applied to differentiable functions, which are functions that are smooth and continuous with no abrupt changes. Some functions, such as step functions or absolute value functions, are not differentiable at certain points.

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