$$\text { Find the formula for } d f ^ { - 1 } / d x \text {. }$$
-$$f ( x ) = 243 x ^ { 5 }$$
A) $\frac { x ^ { 1 / 5 } } { 3 }$
B) $\frac { 1 } { 15 x ^ { 4 / 5 } }$
C) $1215 x ^ { 4 }$
D) $15 x ^ { 4 }$

Question

Quizplus · Accepted Answer

We know that

$$\frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}$$

So, first we need to find $f'(x)$:

$$f(x) = 243x^5$$
$$f'(x) = 1215x^4$$

Now, we need to find $f^{-1}(x)$. We can do this by solving for $x$ in terms of $f(x)$:

$$f(x) = 243x^5$$
$$\frac{f(x)}{243} = x^5$$
$$(\frac{f(x)}{243})^{1/5} = x$$

So, $f^{-1}(x) = (\frac{x}{243})^{1/5}$.

Now, we can substitute into our original formula:

$$\frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))} = \frac{1}{1215[(\frac{x}{243})^{1/5}]^4}$$
$$=\frac{1}{1215}(\frac{243}{x})^{4/5} = \frac{1}{15x^{4/5}}$$

Therefore, the answer is B.

Find the formula for df−1/dx. \text { Find the formula for } d f ^ { - 1 } / d x \text {. } Find the formula for df−1/dx.

$\text { Find the formula for } d f ^ { - 1 } / d x \text {. }$