Differentiate:
f(x) = 2x-(x^2+1)^7 / 3
2-(2x)^6
Would it be 2-x / 3(x^2+1)^6 ?
I get
2 - (x^2+1)^6 (2x)/3
what is this second line? 2-(2x)^6
Well the answer choices are
f'(x) = 2-x / 3(x^2 +1)^6
f'(x)= 2/3 - 14/3x(x^2+1)^6
None of the above
To differentiate the given function f(x) = 2x - (x^2+1)^7 / 3, we'll start by using the quotient rule of differentiation. The quotient rule states that if we have a function in the form f(x) = g(x) / h(x), where g(x) and h(x) are differentiable functions, then the derivative of f(x) is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / h(x)^2
Let's apply the quotient rule to differentiate the given function.
We have g(x) = 2x and h(x) = (x^2+1)^7 / 3.
First, let's find g'(x):
g'(x) = d/dx [2x] = 2
Now, let's find h'(x):
To find h'(x), we need to use the chain rule of differentiation. The chain rule states that if we have a composition of functions, f(g(x)), then the derivative of f(g(x)) is given by:
(f(g(x)))' = f'(g(x)) * g'(x)
In this case, f(u) = u^7 / 3 and g(x) = x^2 + 1.
Using the power rule, we can find f'(u):
f'(u) = d/du [u^7 / 3] = (7u^6) / 3
Now, let's find g'(x):
g'(x) = d/dx [x^2 + 1] = 2x
Now, we can use the chain rule to find h'(x):
h'(x) = f'(g(x)) * g'(x) = (7(g(x))^6 / 3) * (2x) = (7(x^2 + 1)^6 / 3) * (2x)
Now, let's substitute the values of g'(x) and h'(x) into the quotient rule formula:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / h(x)^2
f'(x) = (2 * ((x^2+1)^7 / 3) - 2x * (7(x^2 + 1)^6 / 3) * (2x)) / ((x^2+1)^7 / 3)^2
After simplifying, we get:
f'(x) = (2(x^2+1)^7 - 14x^2(x^2+1)^6) / ((x^2+1)^14 / 9)
Simplifying further, we get:
f'(x) = 2(x^2+1)^7 - 14x^2(x^2+1)^6 / ((x^2+1)^14 / 9)
Therefore, the derivative of f(x) is:
f'(x) = 2(x^2+1)^7 - 14x^2(x^2+1)^6 / ((x^2+1)^14 / 9)
So, the given differentiation is not correct.