Suppose The Slope Of The Curve Y = F ( X Y=f(x Y = F ( X ] At ( 1 , 5 (1,5 ( 1 , 5 ] Is 5 6 \frac{5}{6} 6 5 . Find ( F − 1 ) ′ ( 5 \left(f^{-1}\right)^{\prime}(5 ( F − 1 ) ′ ( 5 ].

Mar 13, 2025 by ADMIN 183 views

**Finding the Derivative of the Inverse Function**

Introduction

In calculus, the concept of inverse functions and their derivatives is crucial for understanding various mathematical concepts. Given a function $f(x)$ , its inverse function $f^{-1}(x)$ is defined as a function that undoes the action of $f(x)$ . In this article, we will explore how to find the derivative of the inverse function $f^{-1}(x)$ , given the derivative of the original function $f(x)$ .

The Relationship Between the Derivative of a Function and its Inverse

To find the derivative of the inverse function $f^{-1}(x)$ , we need to understand the relationship between the derivative of a function and its inverse. Let's consider a function $f(x)$ and its inverse $f^{-1}(x)$ . We know that the derivative of $f(x)$ is denoted as $f'(x)$ , and the derivative of $f^{-1}(x)$ is denoted as $(f^{-1})'(x)$ .

The Formula for the Derivative of the Inverse Function

The formula for the derivative of the inverse function $f^{-1}(x)$ is given by:

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

This formula states that the derivative of the inverse function $f^{-1}(x)$ is equal to the reciprocal of the derivative of the original function $f(x)$ , evaluated at the point $f^{-1}(x)$ .

Applying the Formula to the Given Problem

Now, let's apply the formula to the given problem. We are given that the slope of the curve $y=f(x)$ at $(1,5)$ is $\frac{5}{6}$ . This means that the derivative of the function $f(x)$ at the point $x=1$ is $\frac{5}{6}$ .

Finding the Derivative of the Inverse Function

Using the formula for the derivative of the inverse function, we can find the derivative of $f^{-1}(x)$ at the point $x=5$ . We have:

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

Since $f^{-1}(5) = 1$ , we can substitute this value into the formula:

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

We know that $f'(1) = \frac{5}{6}$ , so we can substitute this value into the formula:

\left(f^{-1}\right)'(5) = \frac{1}{\frac{5}{6}} = \frac{6}{5}

Therefore, the derivative of the inverse function $f^{-1}(x)$ at the point $x=5$ is $\frac{6}{5}$ .

Conclusion

In conclusion, we have seen how to find the derivative of the inverse function $f^{-1}(x)$ , given the derivative of the original function $f(x)$ . We have applied the formula for the derivative of the inverse function to a given problem and found the derivative of the inverse function at a specific point. This concept is crucial for understanding various mathematical concepts, including optimization and physics.

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Calculus, 2nd edition, James Stewart

Glossary

Derivative: A measure of how a function changes as its input changes.
Inverse Function: A function that undoes the action of another function.
Slope: A measure of how steep a curve is at a given point.

FAQs

Q: What is the derivative of the inverse function? A: The derivative of the inverse function is given by the formula: $\left(f^{-1}\right)'(x) = \frac{1}{f'(f^{-1}(x))}$ .
Q: How do I find the derivative of the inverse function? A: To find the derivative of the inverse function, you need to apply the formula for the derivative of the inverse function, using the derivative of the original function and the point at which you want to find the derivative of the inverse function.
Inverse Function Derivative: Frequently Asked Questions =====================================================

Q: What is the inverse function derivative?

A: The inverse function derivative is a measure of how the inverse function changes as its input changes. It is denoted as $(f^{-1})'(x)$ and is used to find the rate of change of the inverse function at a given point.

Q: How do I find the inverse function derivative?

A: To find the inverse function derivative, you need to apply the formula:

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

This formula states that the derivative of the inverse function is equal to the reciprocal of the derivative of the original function, evaluated at the point $f^{-1}(x)$ .

Q: What is the relationship between the derivative of a function and its inverse?

A: The derivative of a function and its inverse are related by the formula:

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

This formula shows that the derivative of the inverse function is equal to the reciprocal of the derivative of the original function, evaluated at the point $f^{-1}(x)$ .

Q: Can I use the chain rule to find the derivative of the inverse function?

A: Yes, you can use the chain rule to find the derivative of the inverse function. The chain rule states that if you have a composite function of the form $f(g(x))$ , then the derivative of the composite function is given by:

(f \circ g)'(x) = f'(g(x)) \cdot g'(x)

You can use this formula to find the derivative of the inverse function by substituting $f^{-1}(x)$ for $g(x)$ and $f(x)$ for $f(g(x))$ .

Q: What is the significance of the inverse function derivative?

A: The inverse function derivative is significant because it allows you to find the rate of change of the inverse function at a given point. This is useful in a variety of applications, including optimization and physics.

Q: Can I use the inverse function derivative to find the derivative of the original function?

A: Yes, you can use the inverse function derivative to find the derivative of the original function. If you know the derivative of the inverse function, you can use the formula:

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

to find the derivative of the original function.

Q: What are some common applications of the inverse function derivative?

A: The inverse function derivative has a variety of applications, including:

Optimization: The inverse function derivative is used to find the maximum or minimum of a function.
Physics: The inverse function derivative is used to describe the motion of objects.
Economics: The inverse function derivative is used to model the behavior of economic systems.

Q: Can I use the inverse function derivative to find the second derivative of the original function?

A: Yes, you can use the inverse function derivative to find the second derivative of the original function. If you know the derivative of the inverse function, you can use the formula:

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

to find the second derivative of the original function.

Q: What are some common mistakes to avoid when using the inverse function derivative?

A: Some common mistakes to avoid when using the inverse function derivative include:

Not using the correct formula for the derivative of the inverse function.
Not evaluating the derivative of the inverse function at the correct point.
Not using the chain rule correctly.

Q: Can I use the inverse function derivative to find the derivative of a multivariable function?

A: Yes, you can use the inverse function derivative to find the derivative of a multivariable function. If you know the derivative of the inverse function, you can use the formula:

\frac{\partial f}{\partial x} = \frac{1}{\frac{\partial f^{-1}}{\partial x} \cdot \frac{\partial f}{\partial f^{-1}}}

to find the derivative of the multivariable function.

Q: What are some common applications of the inverse function derivative in multivariable calculus?

A: The inverse function derivative has a variety of applications in multivariable calculus, including:

Optimization: The inverse function derivative is used to find the maximum or minimum of a multivariable function.
Physics: The inverse function derivative is used to describe the motion of objects in multivariable space.
Economics: The inverse function derivative is used to model the behavior of economic systems in multivariable space.