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Find the derivative of $\left(x^3+2\right)^2\left(x^4+4\right)^4$

Step-by-step Solution

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Final Answer

$6\left(x^3+2\right)x^{2}\left(x^4+4\right)^4+16\left(x^3+2\right)^2\left(x^4+4\right)^{3}x^{3}$
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Step-by-step Solution

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Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\left(x^3+2\right)^2$ and $g=\left(x^4+4\right)^4$

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

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

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Learn how to solve differential calculus problems step by step online. Find the derivative of (x^3+2)^2(x^4+4)^4. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=\left(x^3+2\right)^2 and g=\left(x^4+4\right)^4. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}. The derivative of a sum of two or more functions is the sum of the derivatives of each function.

Final Answer

$6\left(x^3+2\right)x^{2}\left(x^4+4\right)^4+16\left(x^3+2\right)^2\left(x^4+4\right)^{3}x^{3}$

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

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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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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.

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