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The Chain Rule quiz #1

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  • How do you apply the chain rule to find the derivative of the function f(x) = (4x + 5)^3?
    First, identify the outside function (raising to the 3rd power) and the inside function (4x + 5). Take the derivative of the outside function, treating the inside as a variable: 3(4x + 5)^2. Then multiply by the derivative of the inside function, which is 4. The final derivative is 12(4x + 5)^2.
  • How do you use the chain rule multiple times to find the derivative of f(x) = [sin(3x^2)]^4?
    Start with the outermost function (raising to the 4th power): 4[sin(3x^2)]^3. Multiply by the derivative of the next inner function, sin(3x^2), which is cos(3x^2). Then multiply by the derivative of the innermost function, 3x^2, which is 6x. The final derivative is 24x * sin^3(3x^2) * cos(3x^2).
  • What is the first step in applying the chain rule to a composite function?
    Identify the outside function and take its derivative, treating the inside function as a variable.
  • When using the chain rule, what do you do after differentiating the outside function?
    Multiply by the derivative of the inside function.
  • How do you find the derivative of f(x) = (4x + 5)^3 using the chain rule?
    Differentiate the outside (cubed) to get 3(4x + 5)^2, then multiply by the derivative of the inside (4), resulting in 12(4x + 5)^2.
  • How is the chain rule expressed in Leibnitz notation?
    It is written as dydx = dydu * dudx, where y is the outside function and u is the inside function.
  • What is the derivative of f(x) = 2(3x^2 - x)^4 using the chain rule?
    The derivative is 8(3x^2 - x)^3 * (6x - 1).
  • Why might you rewrite a function like sin^4(3x^2) before applying the chain rule?
    Rewriting it as [sin(3x^2)]^4 clarifies the layers of composition, making it easier to apply the chain rule.
  • How do you apply the chain rule multiple times to f(x) = [sin(3x^2)]^4?
    Differentiate the outer power (4[sin(3x^2)]^3), multiply by the derivative of sine (cos(3x^2)), then multiply by the derivative of the innermost function (6x), giving 24x * sin^3(3x^2) * cos(3x^2).
  • What is the general strategy for using the chain rule on functions with several nested layers?
    Start from the outermost function and work inward, multiplying by the derivative of each inner function in turn.