Differentiate. f(x) = (x^2e^x)/(x^2 + e^x)

Name: Problem 23, Chapter 3: Calculus: Early Transcendentals
Uploaded: 2022-09-06T00:42:17-08:00
Duration: 5 min 24 s
Channel: Mary Wakumoto
Description: Differentiate. f(x) = (x^2e^x)/(x^2 + e^x)

Key Concepts

Product Rule

The product rule is a technique applied when differentiating an expression that is the product of two functions. According to this rule, the derivative of a product u(x)v(x) is given by u'(x)v(x) + u(x)v'(x). This rule is frequently used in combination with other differentiation rules to simplify the process of differentiating more complex functions.

Exponential Function Differentiation

Exponential function differentiation involves recognizing that the derivative of an exponential function, such as e^x, is the function itself. This property is a key aspect when dealing with functions that incorporate exponential terms, making it easier to apply differentiation rules efficiently.

Quotient Rule

The quotient rule is used for finding the derivative of a ratio of two differentiable functions. It states that if a function is expressed as a quotient u(x)/v(x), its derivative is given by (u'v - uv')/(v^2). This rule is crucial when dealing with functions that are not products but rather divisions of differentiable functions.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function is changing at any given point. It serves as the basis for determining slopes of curves, rates of change in physical phenomena, and for solving optimization problems.

Transcript

00:01 All right, here we have a fun problem to solve our function in red here at the bottom is x squared, e to the x over x squared plus e to the x.

00:12 And notice that there's a product in the numerator and the whole function itself is a quotient.

00:19 So notice we're going to need to use both the product rule and the quotient rule.

00:24 Just reminder the product rule, if i take the derivative of f times g, then it's a derivative.

00:30 Of f times g plus g prime times f.

00:34 If i do quotient rule instead, it's a derivative of the top times the bottom, nine is derivative of the bottom times the top all over the bottom squared.

00:43 So we're going to use both rules to solve.

00:46 So here we go.

00:48 Okay, so f prime of x is going to be basically a quotient rule.

00:56 And overall, that's my u.

00:59 And that's my v.

01:02 So we're going to first take the derivative of the top.

01:05 Well, the top is itself, and this is going to be a big, long fraction.

01:09 The top is itself a product.

01:12 So we're going to first start off by doing u prime, and that's going to be product rule.

01:16 So derivative of the first function times the second.

01:20 So we get 2x times e to the x plus a derivative of the second, which is just itself, times the first.

01:29 So that's u prime.

01:31 Then we multiply by v and that's just a denominator so we can write it as is.

01:38 Then we have to subtract and we're going to take the derivative of the denominator, derivative of x squared is 2x by power rule, and then a derivative of v the x is itself.

01:51 Then we multiply by the numerator and that's our basically we did this part so far.

02:00 U prime v minus v prime u we still need to divide by the denominator squared so we're going to get over x squared plus e to the x quantity squared.

02:15 All right so this is messy.

02:16 We have completed it, but let's have fun and see if we can clean it up a bit.

02:23 So let's go ahead and distribute the top and see what we get.

02:29 Okay, so f prime of x, i'm going to foil first the top.

02:36 So i'm going to do multiply the first.

02:40 So that gives me 2x a cube, e to the x...

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