Math Problem Statement

Find dy/dx if -y = sin^(-3)(x).

Solution

The given equation is:

y=sin3(x)-y = \sin^{-3}(x)

We need to find dydx\frac{dy}{dx}. To do this, we can differentiate both sides of the equation with respect to xx.

Starting with the given equation:

y=sin3(x)-y = \sin^{-3}(x)

  1. Differentiate both sides with respect to xx:

ddx(y)=ddx(sin3(x))\frac{d}{dx}(-y) = \frac{d}{dx}(\sin^{-3}(x))

  1. Applying the chain rule to the left side:

dydx-\frac{dy}{dx}

  1. Applying the chain rule to the right side:

Let u=sin(x)u = \sin(x), then sin3(x)=u3\sin^{-3}(x) = u^{-3}. Differentiating u3u^{-3} with respect to uu gives:

ddx(sin3(x))=3sin4(x)cos(x)\frac{d}{dx}(\sin^{-3}(x)) = -3\sin^{-4}(x) \cdot \cos(x)

Putting these together, we get:

dydx=3sin4(x)cos(x)-\frac{dy}{dx} = -3\sin^{-4}(x) \cdot \cos(x)

  1. Multiplying both sides by -1 to solve for dydx\frac{dy}{dx}:

dydx=3sin4(x)cos(x)\frac{dy}{dx} = 3\sin^{-4}(x) \cdot \cos(x)

Thus, the derivative dydx\frac{dy}{dx} is:

dydx=3cos(x)sin4(x)\frac{dy}{dx} = 3 \cos(x) \sin^{-4}(x)

Would you like more details on any specific step or have any further questions?

Here are some related questions that might interest you:

  1. How do we use the chain rule in differentiation?
  2. What is the difference between implicit and explicit differentiation?
  3. How can the derivative of a function with a trigonometric term be interpreted graphically?
  4. How would the derivative change if the function was (-y

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Trigonometric Functions

Formulas

Chain Rule: d(u^n)/dx = n * u^(n-1) * du/dx
Trigonometric Derivative: d(sin(x))/dx = cos(x)

Theorems

Chain Rule

Suitable Grade Level

Grades 11-12