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What is the derivative of the inverse function for f(x) = -3x + 3?
What is the derivative of the inverse function for f(x) = -3x + 3?
For the function f(x) = -5x + 1, what is (f^{-1})' (x)?
For the function f(x) = -5x + 1, what is (f^{-1})' (x)?
What is the formula used to find (f^{-1})' (x)?
What is the formula used to find (f^{-1})' (x)?
For the function f(x) = -2x + 2, what is the slope of its inverse at a given x?
For the function f(x) = -2x + 2, what is the slope of its inverse at a given x?
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What is the derivative (f^{−1})' (x) for f(x) = x^7 + x - 3?
What is the derivative (f^{−1})' (x) for f(x) = x^7 + x - 3?
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If f(x) = -4x^3 - 4, what is the derivative of the inverse f^{-1})' (x)?
If f(x) = -4x^3 - 4, what is the derivative of the inverse f^{-1})' (x)?
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What is the significance of the negative sign in the inverse function derivative?
What is the significance of the negative sign in the inverse function derivative?
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For the function f(x) = -2x - 3, what is (f^{-1})' (x)?
For the function f(x) = -2x - 3, what is (f^{-1})' (x)?
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What is the derivative of the function f(x) = -2x + 3?
What is the derivative of the function f(x) = -2x + 3?
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What is the derivative of the function $f(x) = -3x + 3$?
What is the derivative of the function $f(x) = -3x + 3$?
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Using the theorem, what is $(f^{-1})'(x)$ for the function $f(x) = -2x + 3$?
Using the theorem, what is $(f^{-1})'(x)$ for the function $f(x) = -2x + 3$?
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What is the derivative of the function $f(x) = -5x + 1$?
What is the derivative of the function $f(x) = -5x + 1$?
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For the function $f(x) = -2x + 2$, what is its derivative?
For the function $f(x) = -2x + 2$, what is its derivative?
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Using the formula $dy = rac{1}{dx}$ for $y = f(x)$, find $(f^{-1})'(x)$ for $f(x) = -2x - 3$.
Using the formula $dy = rac{1}{dx}$ for $y = f(x)$, find $(f^{-1})'(x)$ for $f(x) = -2x - 3$.
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What is the derivative of the function $f(x) = -4x^3 - 4$?
What is the derivative of the function $f(x) = -4x^3 - 4$?
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For the function $f(x) = x^7 + x - 3$, what is $f'(x)$?
For the function $f(x) = x^7 + x - 3$, what is $f'(x)$?
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What is the derivative of the function $f(x) = 3x^5 + 2x + 5$?
What is the derivative of the function $f(x) = 3x^5 + 2x + 5$?
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Using the inverse derivative theorem, find $(f^{-1})'(x)$ for $f(x) = -4x + 2$.
Using the inverse derivative theorem, find $(f^{-1})'(x)$ for $f(x) = -4x + 2$.
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Study Notes
Derivatives of Inverse Functions
- The derivative of an inverse function can be found directly (by computation) or through a theorem or formula.
Direct Computation
- When finding the derivative of an inverse function directly, you can solve for the inverse function first and then differentiate it.
Theorem (f^-1)'(x) = 1 / f'(f^-1(x))
- This theorem provides a formula that can be used to find the derivative of an inverse function.
- The formula uses the first derivative of the original function at the inverse function, which is f^-1(x).
Inverse Function Formula dy = 1 / (dx / dy)
- This formula establishes a relationship between the derivative of the original function and the derivative of its inverse function.
- You can find the derivative of an inverse function by first expressing the original function in terms of y = f(x), then differentiating both sides with respect to y, and finally expressing the result in terms of x.
Examples
- Examples illustrate the application of different methods to find the derivative of an inverse function.
- The methods include direct computation, using the theorem, and using the inverse function formula.
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Description
This quiz explores the calculation of derivatives of inverse functions. It covers direct computation methods, the relevant theorem, and provides formulas to establish relationships between derivatives of original and inverse functions. Ideal for students studying calculus.
