Can somebody so I how to d/dx((6x^-6)/(1+4x^6)^3/2)
sre. Just use the quotient rule. If
y = u/v where u and v are functions of x, then
y' = (u'v - vu')/v^2
So, in this case, that gives
((-36x^-7)(1+4x^6)^(3/2) - (6x^-6)(3/2 (1+4x^6)^(1/2) (24x^5))/(1+4x^6)^3
You may want to simplify that a bit ...
oops. That is, of course, (u'v - uv')/v^2
(bottom* derivative of top - top * derivative of bottom) / bottom^2
bottom * derivative of top = (1+4x^6)^3/2) (-36 x^-7)
-top*derivative of bottom = -(6x^-6)(3/2)(1+4x^6)^1/2 *(24x^5)
bottom^2 = (1+4x^6)^3
plug and chug
or, if it looks a bit less messy, you can use the product rule. In that case, you have
y = (6/x^6)*(1+4x^6)^(-3/2)
y' = -36/x^7 * (1+4x^6)^(-3/2) + (6/x^6)(-3/2)(1+4x^6)^(-5/2) *(24x^5)
= -36/x^7 (1+4x^6)^(-5/2) (1+19x^6)
check my algebra, of course.
To find the derivative of the function f(x) = (6x^(-6))/(1+4x^6)^(3/2), we can use the quotient rule and the chain rule.
The quotient rule states that if you have the function h(x) = p(x)/q(x), where p(x) and q(x) are functions, then the derivative of h(x) is given by:
h'(x) = (p'(x) * q(x) - p(x) * q'(x))/(q(x))^2.
In this case, p(x) = 6x^(-6) and q(x) = (1+4x^6)^(3/2). To take the derivative, we will need to find the derivatives of p(x) and q(x) first.
Let's start with p(x) = 6x^(-6). To find its derivative, we can use the power rule. The power rule states that if you have a function of the form f(x) = x^n, then its derivative is given by:
f'(x) = n * x^(n-1).
Applying the power rule, we find that p'(x) = -36x^(-7).
Now, let's find the derivative of q(x) = (1+4x^6)^(3/2). We can use the chain rule for this. The chain rule states that if you have a function h(x) = f(g(x)), then its derivative is given by:
h'(x) = f'(g(x)) * g'(x).
In this case, f(u) = u^(3/2), where u = (1+4x^6). Applying the power rule, we have f'(u) = (3/2) * u^(1/2).
To find g'(x), we differentiate g(x) = 1+4x^6. The derivative of a constant (1) with respect to x is zero, and the derivative of 4x^6 is 24x^5. Therefore, g'(x) = 24x^5.
Now, we can apply the quotient rule to find the derivative of f(x). Using the formula:
f'(x) = (p'(x) * q(x) - p(x) * q'(x))/(q(x))^2,
we substitute the values:
f'(x) = (-36x^(-7) * (1+4x^6)^(3/2) - 6x^(-6) * (3/2) * (1+4x^6)^(1/2) * 24x^5) / ((1+4x^6)^(3/2))^2.
Now, we have the derivative of the function f(x). Simplifying the expression further may be possible, but this is the general form of the derivative.