/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Find the derivative \(y^{\prime}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 16

Find the derivative \(y^{\prime}(x)\) implicitly. $$e^{x^{2}} y-3 y=x^{2}+1$$

Short Answer

Expert verified
\[y' = \frac{2x - 2xy e^{x^2}}{e^{x^2}-3}\]

Step by step solution

01

Differentiate the both sides implicitly

The rule of differentiation tells us that the derivative of \(e^{x^2}\) is \(2x e^{x^2}\), while for \(3y\), it's \(3y'\) because y is a function of \(x\). Thus, differentiating the entire equation we get:\((2x e^{x^{2}})y + e^{x^{2}}y' - 3y' = 2x + 0\).
02

Rearrange the equation with \(y'\) on the left side.

Next, we simplify the equation and isolate \(y'\). Move all \(y'\) terms on the left and the remaining terms on the right. This gives:\(y'(e^{x^{2}} - 3) = 2x - (2x e^{x^{2}})y\).
03

Solve for \(y'\).

Finally, \(y'\) is equal to \(\frac{2x - 2xy e^{x^2}}{e^{x^2}-3}\). Ensure the entire equation is simplified correctly to avoid any errors.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative of Exponential Functions
Understanding how to differentiate exponential functions is crucial for solving calculus problems. Let's take the exponential function expressed as e raised to some function of x, written as ef(x). The derivative of this, according to the chain rule, is the exponent's derivative times the original function. In mathematical terms, if you have eu(x), where u(x) is a function of x, the derivative is u'(x)eu(x).

For example, if you have ex2, like in our exercise, the derivative is 2xex2, as x2 has a derivative of 2x. It's essential to remember this pattern when confronted with similar exponential terms, as it frequently appears in calculus problems.
Isolating Derivatives
When solving for derivatives in implicit differentiation, a common step is to isolate the derivative terms on one side of the equation. This is similar to solving for an unknown in algebra. In the exercise, y' represents the derivative of y with respect to x, and we need to find its value.

To isolate y', you begin by getting all terms with y' on one side and everything else on the other. This means breaking apart any products or sums involving y' and treating them like algebraic terms. Once y' is isolated, the rest of the equation consists of functions of x only, allowing us to solve for y' explicitly. This step鈥攎oving from the implicit differentiation to explicitly solving for the derivative鈥攊s a key skill to master.
Differentiation Techniques
There are various techniques to differentiate functions, and using the right one is crucial for finding accurate solutions. The technique of implicit differentiation, used in this problem, is necessary when dealing with variables that are functions of each other, like x and y in the given equation.

Implicit differentiation allows us to take the derivative of both sides of an equation with respect to x, even if it might not be clear how y changes with x. We treat y as an implicit function of x and differentiate as usual, keeping in mind to multiply by y' whenever differentiating a y term. After differentiation, it's necessary to rearrange the equation to solve for the desired derivative, which builds on algebraic manipulation skills.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The given function represents the height of an object. Compute the velocity and acceleration at time \(t=t_{0} .\) Is the object going up or down? Is the speed of the object increasing or decreasing? $$h(t)=-16 t^{2}+40 t+5, t_{0}=2$$

Prove that \(|\sin a|<|a|\) for all \(a \neq 0\) and use the result to show that the only solution to the equation \(\sin x=x\) is \(x=0 .\) What happens if you try to find all intersections with a graphing calculator?

Show that for any real numbers \(u\) and \(v\) \(|\cos u-\cos v| \leq|u-v| .\) (Hint: Use the Mean Value Theorem.)

Use the given position function to find the velocity and acceleration functions. $$s(t)=12 t^{3}-6 t-1$$

Involve the hyperbolic sine and hyperbolic cosine functions: \(\sinh x=\frac{e^{x}-e^{-x}}{2}\) and \(\cosh x=\frac{e^{x}+e^{-x}}{2}\) Show that both \(\sinh x\) and \(\cosh x\) have the property that \(f^{\prime \prime}(x)=f(x)\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.