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Find the derivative of $\frac{x^5-x^4+x^2-2}{x^2+1}$

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Solution

Final Answer

$\frac{3x^{6}+5x^{4}-2x^{5}-4x^{3}+6x}{\left(x^2+1\right)^2}$
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Step-by-step Solution

Problem to solve:

$\frac{d}{dx}\left(\frac{x^5-x^4+x^2-2}{x^2+1}\right)$

Specify the solving method

1

Apply the quotient rule for differentiation, which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)-\left(x^5-x^4+x^2-2\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$
2

Simplify the product $-(x^5-x^4+x^2-2)$

$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)+\left(-x^5-\left(-x^4+x^2-2\right)\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$

Simplify the product $-(-x^4+x^2-2)$

$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)+\left(-x^5-\left(-1\right)x^4-\left(x^2-2\right)\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$

Multiply $-1$ times $-1$

$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)+\left(-x^5+x^4-\left(x^2-2\right)\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$
3

Simplify the product $-(-x^4+x^2-2)$

$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)+\left(-x^5+x^4-\left(x^2-2\right)\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$

Simplify the product $-(x^2-2)$

$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$

Simplify the product $-(-x^4+x^2-2)$

$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)+\left(-x^5-\left(-1\right)x^4-\left(x^2-2\right)\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$

Multiply $-1$ times $-1$

$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)+\left(-x^5+x^4-\left(x^2-2\right)\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$
4

Simplify the product $-(x^2-2)$

$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$
5

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(-x^4\right)+\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-2\right)\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$
6

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(-x^4\right)+\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-2\right)\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(1\right)\right)}{\left(x^2+1\right)^2}$
7

The derivative of the constant function ($-2$) is equal to zero

$\frac{\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(-x^4\right)+\frac{d}{dx}\left(x^2\right)\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(1\right)\right)}{\left(x^2+1\right)^2}$
8

The derivative of the constant function ($1$) is equal to zero

$\frac{\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(-x^4\right)+\frac{d}{dx}\left(x^2\right)\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\frac{d}{dx}\left(x^2\right)}{\left(x^2+1\right)^2}$
9

The derivative of a function multiplied by a constant ($-1$) is equal to the constant times the derivative of the function

$\frac{\left(\frac{d}{dx}\left(x^5\right)-\frac{d}{dx}\left(x^4\right)+\frac{d}{dx}\left(x^2\right)\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\frac{d}{dx}\left(x^2\right)}{\left(x^2+1\right)^2}$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$5x^{\left(5-1\right)}$

Subtract the values $5$ and $-1$

$5x^{4}$
10

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{\left(5x^{4}-\frac{d}{dx}\left(x^4\right)+\frac{d}{dx}\left(x^2\right)\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\frac{d}{dx}\left(x^2\right)}{\left(x^2+1\right)^2}$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$-4x^{\left(4-1\right)}$

Subtract the values $4$ and $-1$

$-4x^{3}$
11

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{\left(5x^{4}-4x^{3}+\frac{d}{dx}\left(x^2\right)\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\frac{d}{dx}\left(x^2\right)}{\left(x^2+1\right)^2}$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$2x^{\left(2-1\right)}$

Subtract the values $2$ and $-1$

$2x$
12

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{\left(5x^{4}-4x^{3}+2x\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\frac{d}{dx}\left(x^2\right)}{\left(x^2+1\right)^2}$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$2\left(-x^5+x^4-x^2+2\right)x^{\left(2-1\right)}$

Subtract the values $2$ and $-1$

$2\left(-x^5+x^4-x^2+2\right)x$
13

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{\left(5x^{4}-4x^{3}+2x\right)\left(x^2+1\right)+2\left(-x^5+x^4-x^2+2\right)x}{\left(x^2+1\right)^2}$

Multiply the single term $x^2+1$ by each term of the polynomial $\left(5x^{4}-4x^{3}+2x\right)$

$5x^{4}\left(x^2+1\right)-4x^{3}\left(x^2+1\right)+2x\left(x^2+1\right)+2\left(-x^5+x^4-x^2+2\right)x$

Multiply the single term $5x^{4}$ by each term of the polynomial $\left(x^2+1\right)$

$5x^2x^{4}+5x^{4}-4x^{3}\left(x^2+1\right)+2x\left(x^2+1\right)+2\left(-x^5+x^4-x^2+2\right)x$

When multiplying exponents with same base we can add the exponents

$5x^{6}+5x^{4}-4x^{3}\left(x^2+1\right)+2x\left(x^2+1\right)+2\left(-x^5+x^4-x^2+2\right)x$

Multiply the single term $-4x^{3}$ by each term of the polynomial $\left(x^2+1\right)$

$5x^{6}+5x^{4}-4x^2x^{3}-4x^{3}+2x\left(x^2+1\right)+2\left(-x^5+x^4-x^2+2\right)x$

When multiplying exponents with same base we can add the exponents

$5x^{6}+5x^{4}-4x^{5}-4x^{3}+2x\left(x^2+1\right)+2\left(-x^5+x^4-x^2+2\right)x$

Multiply the single term $2x$ by each term of the polynomial $\left(x^2+1\right)$

$5x^{6}+5x^{4}-4x^{5}-4x^{3}+2x^2x+2x+2\left(-x^5+x^4-x^2+2\right)x$

When multiplying exponents with same base you can add the exponents: $2x^2x$

$5x^{6}+5x^{4}-4x^{5}-4x^{3}+2x^{3}+2x+2\left(-x^5+x^4-x^2+2\right)x$

Multiply the single term $2x$ by each term of the polynomial $\left(-x^5+x^4-x^2+2\right)$

$5x^{6}+5x^{4}-4x^{5}-4x^{3}+2x^{3}+2x-2x^5x+2x^4x-2x^2x+4x$

When multiplying exponents with same base you can add the exponents: $-2x^5x$

$5x^{6}+5x^{4}-4x^{5}-4x^{3}+2x^{3}+2x-2x^{6}+2x^4x-2x^2x+4x$

When multiplying exponents with same base you can add the exponents: $2x^4x$

$5x^{6}+5x^{4}-4x^{5}-4x^{3}+2x^{3}+2x-2x^{6}+2x^{5}-2x^2x+4x$

When multiplying exponents with same base you can add the exponents: $-2x^2x$

$5x^{6}+5x^{4}-4x^{5}-4x^{3}+2x^{3}+2x-2x^{6}+2x^{5}-2x^{3}+4x$

Combining like terms $5x^{6}$ and $-2x^{6}$

$3x^{6}+5x^{4}-4x^{5}-4x^{3}+2x^{3}+2x+2x^{5}-2x^{3}+4x$

Combining like terms $-4x^{5}$ and $2x^{5}$

$3x^{6}+5x^{4}-2x^{5}-4x^{3}+2x^{3}+2x-2x^{3}+4x$

Combining like terms $-4x^{3}$ and $2x^{3}$

$3x^{6}+5x^{4}-2x^{5}-2x^{3}+2x-2x^{3}+4x$

Combining like terms $-2x^{3}$ and $-2x^{3}$

$3x^{6}+5x^{4}-2x^{5}-4x^{3}+2x+4x$

Combining like terms $2x$ and $4x$

$3x^{6}+5x^{4}-2x^{5}-4x^{3}+6x$
14

Simplify the derivative

$\frac{3x^{6}+5x^{4}-2x^{5}-4x^{3}+6x}{\left(x^2+1\right)^2}$

Final Answer

$\frac{3x^{6}+5x^{4}-2x^{5}-4x^{3}+6x}{\left(x^2+1\right)^2}$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind derivative of (x^5+-1x^4)/(x^2+1) using the product ruleFind derivative of (x^5+-1x^4)/(x^2+1) using the quotient ruleFind derivative of (x^5+-1x^4)/(x^2+1) using logarithmic differentiationFind derivative of (x^5+-1x^4)/(x^2+1) using the definition

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asinh
acosh
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acoth
asech
acsch

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