Math / Calculus

QuestionScore Part III Find the $\frac{d y}{d x}$ . ( 4 scores per question. The total is 12 scores.)
12. $y=x \sin x$ .
13. $y=\frac{x^{2}-1}{x^{2}+1}$ .

Studdy Solution

STEP 1

Assumptions
1. We are given two functions for which we need to find the derivative $\frac{dy}{dx}$ .
2. The first function is $y = x \sin x$ .
3. The second function is $y = \frac{x^2 - 1}{x^2 + 1}$ .
4. We will use differentiation rules such as the product rule and the quotient rule.

STEP 2

For the first function $y = x \sin x$ , we will use the product rule for differentiation. The product rule states that if $y = u \cdot v$ , then $\frac{dy}{dx} = u'v + uv'$ .

STEP 3

Identify $u$ and $v$ for the function $y = x \sin x$ .
Let $u = x$ and $v = \sin x$ .

STEP 4

Differentiate $u = x$ with respect to $x$ .
$u' = \frac{d}{dx}(x) = 1$

STEP 5

Differentiate $v = \sin x$ with respect to $x$ .
$v' = \frac{d}{dx}(\sin x) = \cos x$

STEP 6

Apply the product rule to find $\frac{dy}{dx}$ for $y = x \sin x$ .
$\frac{dy}{dx} = u'v + uv' = 1 \cdot \sin x + x \cdot \cos x$

STEP 7

Simplify the expression for $\frac{dy}{dx}$ .
$\frac{dy}{dx} = \sin x + x \cos x$

STEP 8

Now, move on to the second function $y = \frac{x^2 - 1}{x^2 + 1}$ . We will use the quotient rule for differentiation. The quotient rule states that if $y = \frac{u}{v}$ , then $\frac{dy}{dx} = \frac{u'v - uv'}{v^2}$ .

STEP 9

Identify $u$ and $v$ for the function $y = \frac{x^2 - 1}{x^2 + 1}$ .
Let $u = x^2 - 1$ and $v = x^2 + 1$ .

STEP 10

Differentiate $u = x^2 - 1$ with respect to $x$ .
$u' = \frac{d}{dx}(x^2 - 1) = 2x$

STEP 11

Differentiate $v = x^2 + 1$ with respect to $x$ .
$v' = \frac{d}{dx}(x^2 + 1) = 2x$

STEP 12

Apply the quotient rule to find $\frac{dy}{dx}$ for $y = \frac{x^2 - 1}{x^2 + 1}$ .
$\frac{dy}{dx} = \frac{u'v - uv'}{v^2} = \frac{(2x)(x^2 + 1) - (x^2 - 1)(2x)}{(x^2 + 1)^2}$

STEP 13

Simplify the expression for $\frac{dy}{dx}$ .
First, expand the terms in the numerator:
$(2x)(x^2 + 1) = 2x^3 + 2x$ $(x^2 - 1)(2x) = 2x^3 - 2x$

STEP 14

Subtract the second expression from the first in the numerator:
$2x^3 + 2x - (2x^3 - 2x) = 2x^3 + 2x - 2x^3 + 2x = 4x$

STEP 15

Write the simplified expression for $\frac{dy}{dx}$ .
$\frac{dy}{dx} = \frac{4x}{(x^2 + 1)^2}$

STEP 16

The derivatives for the given functions are:
For $y = x \sin x$ , $\frac{dy}{dx} = \sin x + x \cos x$ .
For $y = \frac{x^2 - 1}{x^2 + 1}$ , $\frac{dy}{dx} = \frac{4x}{(x^2 + 1)^2}$ .

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