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Find the derivative of $\left(x^2+2\right)^3\left(1-x^3\right)^4$

Step-by-step Solution

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Final Answer

$6\left(x^2+2\right)^{2}x\left(1-x^3\right)^4-12\left(x^2+2\right)^3\left(1-x^3\right)^{3}x^{2}$
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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 $f=\left(x^2+2\right)^3$ and $g=\left(1-x^3\right)^4$

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

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

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Learn how to solve integrals of exponential functions problems step by step online. Find the derivative of (x^2+2)^3(1-x^3)^4. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=\left(x^2+2\right)^3 and g=\left(1-x^3\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

$6\left(x^2+2\right)^{2}x\left(1-x^3\right)^4-12\left(x^2+2\right)^3\left(1-x^3\right)^{3}x^{2}$

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^2+2)^3(1+-1x^3)^4 using the product ruleFind derivative of (x^2+2)^3(1+-1x^3)^4 using the quotient ruleFind derivative of (x^2+2)^3(1+-1x^3)^4 using logarithmic differentiationFind derivative of (x^2+2)^3(1+-1x^3)^4 using the definition

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Function Plot

Plotting: $6\left(x^2+2\right)^{2}x\left(1-x^3\right)^4-12\left(x^2+2\right)^3\left(1-x^3\right)^{3}x^{2}$

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5
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7
8
9
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

How to improve your answer:

Main Topic: Integrals of Exponential Functions

Those are integrals that involve exponential functions. Recall that an exponential function is a function of the form f(x)=a^x.

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