Mathematics: What is dy/dx if y = (x^2 + 2)^3 (x^3 + 3)^2and explain how you got your answer. i have no knowledge on how to get the answer at all.

3.875or3 7/8Step-by-step explanation:...

What is dy/dx if y = (x^2 + 2)^3 (x^3 + 3)^2and - id2233321045961, pgjohnston001

Ответ:

ant5784tgi

13.03.2022

Mathematics

We're given the following equation:
y=(x^2+2)^3(x^3+3)^2

In order to find $\frac{dy}{dx}$ we must differentiate both sides of the equation.

Lets start differentiating the left side (

):
$\frac{d}{dy} y=(1)dy=dy$

The $\frac{d}{dy}$ simply servers to let us know we're differentiating whatever follows it (in this case

) with respect to

.

What we used to get the result is called the "power rule for differentiation" it states the following:
$\frac{d}{ds} s^{n}=ns^{n-1}$

In which

is any variable (in the previous case

) and

is any constant (in the previous case this n=1

).

Now we'll differentiate the right side of the equation ( (x^2+2)^3(x^3+3)^2

):
$\frac{d}{dx}(x^2+2)^3(x^3+3)^2=[6x(x^2+2)^2(x^3+3)^2+6x^2(x^3+3)(x^2+2)^3]dx$

What we did to differentiate the right side was, first, apply something called "product rule" for differentiation, it states the following:
$\frac{d}{ds}[f(s)g(s)]= \frac{d}{ds}[f(s)]g(s)+\frac{d}{ds}[g(s)]f(s)$

In which f(s)

and

are arbitrary functions of an arbitrary variable (

) (in this case f(s)=f(x)=(x^2+2)^3

and

).

After that we applied something called "chain rule" for differentiation, which states the following:
if h(s)=g(f(s))

, then
$\frac{d}{ds}[h(s)]= \frac{d}{ds}[g(f(s))] \frac{d}{ds}[f(s)]$

Finally, the

we introduced as a factor after differentiating the right side (we also did it with the left side but with a

) is a consequence of the chain rule, it is always done.

Finally, equaling both differentiated sides of the equation we have:
dy=[6x(x^2+2)^2(x^3+3)^2+6x^2(x^3+3)(x^2+2)^3]dx

dy=[6x(x^2+2)^2(x^3+3)^2+6x^2(x^3+3)(x^2+2)^3]dx

We solve for $\frac{dy}{dx}$ , and the answer is:
$\frac{dy}{dx} =6x(x^2+2)^2(x^3+3)^2+6x^2(x^3+3)(x^2+2)^3$

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What is dy/dx if y = (x^2 + 2)^3 (x^3 + 3)^2and explain how you got your answer. i have no knowledge on how to get the answer at all.

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