For the function f(x)=5root(5)(x+10), find f^(-1)(x). f^(-1)(x)=(x^(5))/(5)-10 f^(-1)(x)=((x)/(5))^(5)-10 f^(-1)(x)=((x)/(5)-10)^(5) f^(-1)(x)=((x-10)^(5))/(5)

Replace with y: Replace f(x) with y. y = 5√(x+10)Swap x and y: Swap x and y. x = 5√(y+10)Solve for y: Solve for y. To isolate y, we first divide both sides by 5: x/5 = √(y+10)Square to eliminate root: Square both sides to eliminate the square root. (x/5)^2 = y + 10Subtract 10 for y: Subtract 10 from both sides to solve for y. y = (x/5)^2 - 10Replace with f^(-1)(x): Replace y with f^(-1)(x) to express the inverse function. f^(-1)(x) = (x/5)^2 - 10

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Find derivatives of using multiple formulae

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Q. For the function

f(x)=5 \sqrt[5]{x+10}

, find

f^{-1}(x)

\newline

f^{-1}(x)=\frac{x^{5}}{5}-10

\newline

f^{-1}(x)=\left(\frac{x}{5}\right)^{5}-10

\newline

f^{-1}(x)=\left(\frac{x}{5}-10\right)^{5}

\newline

f^{-1}(x)=\frac{(x-10)^{5}}{5}

Replace with $y$ : Replace $f(x)$ with $y$ . $\newline$ $y = 5\sqrt{x+10}$
Swap x and y: Swap x and y. $\newline$ $x = 5\sqrt{y+10}$
Solve for y: Solve for y. $\newline$ To isolate y, we first divide both sides by $5$ : $\newline$ $\frac{x}{5} = \sqrt{y+10}$
Square to eliminate root: Square both sides to eliminate the square root. $\newline$ $(\frac{x}{5})^2 = y + 10$
Subtract $10$ for $y$ : Subtract $10$ from both sides to solve for $y$ . $y = \left(\frac{x}{5}\right)^2 - 10$
Replace with $f^{-1}(x)$ : Replace $y$ with $f^{-1}(x)$ to express the inverse function. $\newline$ $f^{-1}(x) = \left(\frac{x}{5}\right)^2 - 10$