Mathematics, 18.07.2019 16:00 simegustachocolate

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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Answer from: momneedshelphmwk

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

Answer from: Quest

the largest number of the four options is 1.8 x 10-2

you have it in scientific notation.

1.80 x 10-6 = 0.00000180

1.80 x 10-4 = 0.000180 (bigger, or "less small", than the former one)

1.8 x 10-2 = 0.0180. as you see. the "less small", or the biggest one of them all.

as you have negative exponent, it means you are dividing 1.8 by 100 (i.e. moving the dot towards left, two times).

if you had positive exponent for the scientific notation, power of tenths, you would be multiplying times 100 (moving the dot towards right) as below:

1.8 x (10)2 = 180

so. if you have negative exponent in scientific notation, move the dot toward left. if it is positive, toward right as many times the exponent tells you.

Answer from: Quest

sure i'm up for it

step-by-step explanation:

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