Hard Trignometric Derivative Problem

In summary: Added in Edit:You can also use the product rule if you want. Just replace the sine with the product and you're good to go.
  • #1
fernanhen
10
0

Homework Statement



Find the derivative of sin(tan(square root of sinx))


Homework Equations



derivative of:

sin=cos
tan=sec squared
sinx=cosx

The Attempt at a Solution



cos(tan(square root of sinx))(sin(sec^2(1/2sinx)^-1/2(cosx))

So I did the derivative of the sin, left what's inside the parenthesis alone.

Then left sin alone, and did the chain rule inside. I'm not sure if I should also have used the product rule as well.

Help:cry:
 
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  • #2
fernanhen said:

Homework Statement



Find the derivative of sin(tan(square root of sinx))

Homework Equations



derivative of:

sin=cos
tan=sec squared
sinx=cosx

The Attempt at a Solution



cos(tan(square root of sinx))(sin(sec^2(1/2sinx)^-1/2(cosx))

So I did the derivative of the sin, left what's inside the parenthesis alone.

[STRIKE]Then left sin alone[/STRIKE], and did the chain rule inside. I'm not sure if I should also have used the product rule as well.

Help:cry:
Drop the sine that I crossed out. There are other errors too.

[itex]\displaystyle \frac{d}{dx}f\left(g\left(h(x)\right)\right)= f'\left(g\left(h(x)\right)\right)\cdot g'\left(h(x)\right)\cdot h'(x)[/itex]

Added in Edit:
There are four functions nested.

Therefore, [itex]\displaystyle \frac{d}{dx}w\left(f\left(g\left(h(x)\right)\right)\right)= w'\left(f\left(g\left(h(x)\right)\right)\right) \cdot f'\left(g\left(h(x)\right)\right)\cdot g'\left(h(x)\right)\cdot h'(x)[/itex]
 
Last edited:
  • #3
Nope, you don't need the product rule, because there is no multiplication of anything by anything else here. There are no products of functions. You have a composition of functions (i.e. a function of a function etc), which means that the chain rule is exactly what you need. To help keep track of everything, it might be useful to do some substitutions. Start with the innermost function and work your way out.

Let y = sin x

let u = √y

let v = tan u

let w = sin v

So we have a composition of functions since w = w(v) = w( v(u) ) = w( v( u(y) ) )

= w( v( u( y(x) ) ) )

It's function of a function of a function of a function. So the chain rule says that:$$\frac{dw}{dx} = \frac{dw}{dv}\frac{dv}{du}\frac{du}{dy}\frac{dy}{dx}$$So all you have to do is evaluate these four derivatives separately and then multiply them together.
 
Last edited:
  • #4
fernanhen said:

Homework Statement



Find the derivative of sin(tan(square root of sinx))


Homework Equations



derivative of:

sin=cos
tan=sec squared
sinx=cosx

The Attempt at a Solution



cos(tan(square root of sinx))(sin(sec^2(1/2sinx)^-1/2(cosx))

So I did the derivative of the sin, left what's inside the parenthesis alone.

Then left sin alone, and did the chain rule inside. I'm not sure if I should also have used the product rule as well.

Help:cry:

Just do the multi-function chain rule (is that what its called?). Not the multi-variable chain rule, the multifunction chain rule.

dy/dt=dy/df df/dg dg/dh ... dp/dt
 

FAQ: Hard Trignometric Derivative Problem

1. What is a hard trigonometric derivative problem?

A hard trigonometric derivative problem is a calculus problem that involves finding the derivative of a function that contains trigonometric functions, such as sine, cosine, or tangent. These problems can be challenging because they require knowledge of trigonometric identities and the chain rule.

2. How do I solve a hard trigonometric derivative problem?

To solve a hard trigonometric derivative problem, you will need to use trigonometric identities to simplify the function, and then apply the chain rule to find the derivative. It is also helpful to have a good understanding of how trigonometric functions behave and their derivatives.

3. What are some common trigonometric identities used in solving derivative problems?

Some common trigonometric identities used in solving derivative problems include the power reduction formula, double angle formula, half angle formula, and sum and difference formulas.

4. Can I use a calculator to solve a hard trigonometric derivative problem?

While a calculator can be helpful in checking your work or simplifying the problem, it is important to have a good understanding of the concepts and how to solve the problem by hand. Relying solely on a calculator can lead to errors and a lack of understanding.

5. What are some tips for solving hard trigonometric derivative problems?

Some tips for solving hard trigonometric derivative problems include practicing with different types of problems, using trigonometric identities to simplify the function, and breaking the problem down into smaller steps. It is also important to carefully check your work and understand the steps you took to solve the problem.

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