Find the derivative of $\frac{x^5-6x^4-52x^3+218x^2+819x-980}{x^3+8x^2+11x-20}$

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Final answer to the problem

$\frac{\left(5x^{4}-24x^{3}-156x^{2}+436x+819\right)\left(x^3+8x^2+11x-20\right)+\left(-x^5+6x^4+52x^3-218x^2-819x+980\right)\left(3x^{2}+16x+11\right)}{\left(x^3+8x^2+11x-20\right)^2}$
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Step-by-step Solution

How should I solve this problem?

  • Find the derivative
  • Find the derivative using the definition
  • Find the derivative using the product rule
  • Find the derivative using the quotient rule
  • Find the derivative using logarithmic differentiation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Integrate by parts
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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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{\frac{d}{dx}\left(x^5-6x^4-52x^3+218x^2+819x-980\right)\left(x^3+8x^2+11x-20\right)-\left(x^5-6x^4-52x^3+218x^2+819x-980\right)\frac{d}{dx}\left(x^3+8x^2+11x-20\right)}{\left(x^3+8x^2+11x-20\right)^2}$

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$\frac{\frac{d}{dx}\left(x^5-6x^4-52x^3+218x^2+819x-980\right)\left(x^3+8x^2+11x-20\right)-\left(x^5-6x^4-52x^3+218x^2+819x-980\right)\frac{d}{dx}\left(x^3+8x^2+11x-20\right)}{\left(x^3+8x^2+11x-20\right)^2}$

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Learn how to solve differential calculus problems step by step online. Find the derivative of (x^5-6x^4-52x^3218x^2819x+-980)/(x^3+8x^211x+-20). 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}}. Simplify the product -(x^5-6x^4-52x^3+218x^2+819x-980). Simplify the product -(-6x^4-52x^3+218x^2+819x-980). Simplify the product -(-52x^3+218x^2+819x-980).

Final answer to the problem

$\frac{\left(5x^{4}-24x^{3}-156x^{2}+436x+819\right)\left(x^3+8x^2+11x-20\right)+\left(-x^5+6x^4+52x^3-218x^2-819x+980\right)\left(3x^{2}+16x+11\right)}{\left(x^3+8x^2+11x-20\right)^2}$

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Function Plot

Plotting: $\frac{\left(5x^{4}-24x^{3}-156x^{2}+436x+819\right)\left(x^3+8x^2+11x-20\right)+\left(-x^5+6x^4+52x^3-218x^2-819x+980\right)\left(3x^{2}+16x+11\right)}{\left(x^3+8x^2+11x-20\right)^2}$

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x
y
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.
(◻)
+
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×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

How to improve your answer:

Main Topic: Differential Calculus

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.

Used Formulas

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