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Find the derivative $\frac{d}{dx}\left(\frac{2x^3+5}{4x^2+7}\right)$

Step-by-step Solution

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Final Answer

$\frac{8x^{4}+42x^{2}-40x}{\left(4x^2+7\right)^2}$
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Step-by-step Solution

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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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{\frac{d}{dx}\left(2x^3+5\right)\left(4x^2+7\right)-\left(2x^3+5\right)\frac{d}{dx}\left(4x^2+7\right)}{\left(4x^2+7\right)^2}$

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$\frac{\frac{d}{dx}\left(2x^3+5\right)\left(4x^2+7\right)-\left(2x^3+5\right)\frac{d}{dx}\left(4x^2+7\right)}{\left(4x^2+7\right)^2}$

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Learn how to solve quotient rule of differentiation problems step by step online. Find the derivative d/dx((2x^3+5)/(4x^2+7)). 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 -(2x^3+5). The derivative of a sum of two or more functions is the sum of the derivatives of each function. The derivative of a sum of two or more functions is the sum of the derivatives of each function.

Final Answer

$\frac{8x^{4}+42x^{2}-40x}{\left(4x^2+7\right)^2}$

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 (2x^3+5)/(4x^2+7) using the product ruleFind derivative of (2x^3+5)/(4x^2+7) using the quotient ruleFind derivative of (2x^3+5)/(4x^2+7) using logarithmic differentiationFind derivative of (2x^3+5)/(4x^2+7) using the definition

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Plotting: $\frac{8x^{4}+42x^{2}-40x}{\left(4x^2+7\right)^2}$

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0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
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

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