# What is the derivative of #sinx^tanx#?

[Assuming you meant

We have:

We use a rule you may be unfamiliar with:

Therefore, we have:

The product rule:

Remember that:

Another thing to remember here:

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[Assuming you meant

We have:

Here is a rule that you may not be familiar with:

Use the product rule:

Remember that:

By signing up, you agree to our Terms of Service and Privacy Policy

To find the derivative of ( \sin(x^{\tan(x)}) ), you need to use the chain rule.

Let ( u = x^{\tan(x)} ).

Then, ( \frac{du}{dx} = \tan(x) \cdot x^{\tan(x) - 1} + \ln(x) \cdot x^{\tan(x)} \cdot \sec^2(x) ).

Now, differentiate ( \sin(u) ) with respect to ( u ), which is ( \cos(u) ).

Finally, multiply ( \cos(u) ) by ( \frac{du}{dx} ) to get the derivative with respect to ( x ).

The derivative of ( \sin(x^{\tan(x)}) ) with respect to ( x ) is:

[ \cos(x^{\tan(x)}) \left( \tan(x) \cdot x^{\tan(x) - 1} + \ln(x) \cdot x^{\tan(x)} \cdot \sec^2(x) \right) ]

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