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(a) Show that $ f(x) = x + r^x $ is one-to-one.(b) What is the value of $ f^{-1}(1)? $(c) Use the formula from Exercise 77(a) to find $ (f^{-1})'(1). $

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00:59

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Campbell University

Oregon State University

Harvey Mudd College

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

00:26

Compute $f^{\prime}(a)$ al…

01:23

Find a formula for $f^{-1}…

03:40

Use the formula in Exercis…

01:07

01:51

Find $f^{-1}.$$$f(x)=\…

in this problem, we're given a function of black sort of perfect explosive. Each X in front 81 short at this punch. One hole. Let's say, uh, look at first, they were job dysfunction. That final thanks. That is one plus X. Well, let's look at the second derivative. And this each of the eggs 37 10 would be the JD X, and you would get the same behavior for all fire order. There it is. So for any value of X, we know that each of the eggs will be positive, meaning that all the derivatives will be positive. So it means that dysfunction is increasing always since that is always increasing. Increasing this function is 12 or all right park years front. Yours or no, let's say that, um oh, um, that's is so f universe. Let's write it this time instead of f universe off X plus, each of eggs is equal. Thanks. We want an X plus, you two x to be one. If that is born, we see that X should zero, and that is the answer. So we consider a f in worse off one is unequalled zero. All right, import CVS, But English front or beginning used the question that we drive in problem 77 Soviet. Oh, that f inwards Prime off exit was born over f crime of immerse effects. So that is more over a probable and universal one in part. Maybe find FT merciful as zero. So that is warm or prime. Zero. What? Is that a problem? Zero. We know that prime effects from A is equal to one plus you to the ex that primary zero would be born boss Jiro And that is due so that the answer would be, um, dinner's prime one. Is it on over to?

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