# How do you find the derivative of Inverse trig function #y = (sin(3x) + cot(x^3))^8#?

From the looks of it, you're going to have to use the chain rule three times to differentiate this function.

Before you get started, keep in mind that

This means that you can write

and

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

To find the derivative of the given function (y = (\sin(3x) + \cot(x^3))^8), we will use the chain rule, the product rule, and the derivatives of basic trigonometric functions. The chain rule is used when differentiating a composite function, which in this case is a function raised to a power. The product rule may be needed for differentiating expressions like (\cot(x)), which is equivalent to (\frac{\cos(x)}{\sin(x)}).

Given: [y = (\sin(3x) + \cot(x^3))^8]

Let (u = \sin(3x) + \cot(x^3)), so that (y = u^8).

First, differentiate (y) with respect to (u) (using the power rule): [\frac{dy}{du} = 8u^7]

Next, differentiate (u) with respect to (x): [u = \sin(3x) + \cot(x^3)] [du/dx = \frac{d}{dx}[\sin(3x)] + \frac{d}{dx}[\cot(x^3)]]

The derivative of (\sin(3x)) with respect to (x), using the chain rule, is: [3\cos(3x)]

The derivative of (\cot(x^3)) with respect to (x), recognizing that (\cot(x) = \frac{1}{\tan(x)}) or (\cos(x)/\sin(x)), and applying the chain rule, is: [-\csc^2(x^3) \cdot 3x^2] Which simplifies to: [-3x^2\csc^2(x^3)]

Thus: [du/dx = 3\cos(3x) - 3x^2\csc^2(x^3)]

Finally, apply the chain rule by multiplying (\frac{dy}{du}) and (\frac{du}{dx}) to find (\frac{dy}{dx}): [\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 8(\sin(3x) + \cot(x^3))^7 \cdot (3\cos(3x) - 3x^2\csc^2(x^3))]

Therefore, the derivative of the given function with respect to (x) is: [\frac{dy}{dx} = 8(\sin(3x) + \cot(x^3))^7 \cdot (3\cos(3x) - 3x^2\csc^2(x^3))]

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