# How do you differentiate #y = x^(-sin(x) )#?

Using the fact that

We let

So

The formula we've used is

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

To differentiate ( y = x^{-\sin(x)} ), we use the chain rule.

Given ( y = x^{-\sin(x)} ), we first take the natural logarithm of both sides:

[ \ln(y) = -\sin(x) \cdot \ln(x) ]

Then, we differentiate both sides with respect to ( x ):

[ \frac{1}{y} \cdot \frac{dy}{dx} = -\cos(x) \cdot \ln(x) - \frac{\sin(x)}{x} ]

Finally, we solve for ( \frac{dy}{dx} ):

[ \frac{dy}{dx} = y \cdot \left( -\cos(x) \cdot \ln(x) - \frac{\sin(x)}{x} \right) ]

Substituting back ( y = x^{-\sin(x)} ):

[ \frac{dy}{dx} = x^{-\sin(x)} \cdot \left( -\cos(x) \cdot \ln(x) - \frac{\sin(x)}{x} \right) ]

This is the derivative of ( y = x^{-\sin(x)} ).

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

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7