f(x)= (x^2-1)^3,is its derivative f'(x)= 6x^5 -12x^3 + 6x.
Is there an alternative answer, because a computer generated answer is diff then mine. please help.
yes, that's the derivative.
did you expand your answer? or how did you do it?
I used the chain rule... my working is above
y = u^3 where u= (x^2-1)
so dy/du = 3u^2
and du/dx = 2x
then dy/dx = 3u^2(2x)
so after that substitute u
3(x^2-1)^2(2x)
=6x(x^2-1)^2
and I expanded it and got
6x^5-12x^3+6x
is that correct?
because a computer generated answer says you can simplify (x^2-1) since it is a perfect square and they get a diff answer than mine? However i cannot understand their simplification.
yes, your answer is correct. :)
i think that simplification means to factor 6x(x^2-1)^2 into 6x[(x-1)^2][(x+1)^2]
I majorly confused here, I am using a computer software which gives me a different answer and i am doubting my work over an over again, the computer software tells me:
"squaring the expression means multiplying the expression by itself"
and then they expand 6x[(x-1)(x-1)][x+1]^2
and in the next step then suddenly (x-1) is canceled and they just consider (x+1)(x+1)
Can someone explain me why the software does that, or is it just one of their ways to trick its users?
To find the derivative of the function f(x) = (x^2 - 1)^3, let's apply the chain rule. The chain rule states that if we have a function g(x) raised to a power n, then its derivative can be found by multiplying the derivative of g(x) by n*g^(n-1)(x), where g' is the derivative of g.
First, let's find the derivative of g(x) = x^2 - 1:
g'(x) = 2x
Now, applying the chain rule, the derivative of f(x) = (x^2 - 1)^3 will be:
f'(x) = 3(x^2 - 1)^2 * (2x)
Expanding this expression, we get:
f'(x) = 6x(x^2 - 1)^2
However, this is not the same as the answer you provided as f'(x) = 6x^5 - 12x^3 + 6x. It seems like there might be an error in either your or the computer-generated answer.
To double-check the computer-generated answer, you can manually calculate the derivative using the power rule and simplify to see if it matches the given answer:
Starting with f(x) = (x^2 - 1)^3, let's find f'(x) using the power rule:
f'(x) = 3(x^2 - 1)^2 * d/dx(x^2 - 1)
Using the power rule on the term (x^2 - 1), we have:
d/dx(x^2 - 1) = 2x
Substituting this back into the original equation, we get:
f'(x) = 3(x^2 - 1)^2 * 2x
Expanding further, we get:
f'(x) = 6x(x^2 - 1)^2
Thus, the computer-generated answer, f'(x) = 6x^5 - 12x^3 + 6x, seems to be incorrect. The correct derivative should be f'(x) = 6x(x^2 - 1)^2.