i need to find the derivative using chain rule and show steps:
y= (x) / ((x^2)-1)^(-1/2)
and is it possible to have free tutoring online like through a chatroom? i am in need of help
there are tutoring chatrooms, but not here.
To play around with derivatives and much more, visit wolframalpha.com. For this problem, enter
derivative (x) / ((x^2)-1)^(-1/2)
and it will show you
(2x^2-1)/√(x^2-1)
and you can click on the "step-by-step solution" button for the details
y= (x) / ((x^2)-1)^(-1/2)
do you have a typo? That is the same as:
y = x (x^2-1)^(+1/2)
dy/dx = x(1/2)(x^2-1)^-.5(2x) +(x^2-1)^.5
= x^2(x^2-1)^-.5 + (x^2-1)^.5
To find the derivative of the given function using the chain rule, you can follow these steps:
Step 1: Identify the function parts
Split the function into smaller functions where you have an "outer" function and an "inner" function. In this case, the outer function is y = u/v, and the inner function is u = x and v = (x^2 - 1)^(-1/2).
Step 2: Find the derivatives of the inner functions
Take the derivatives of the inner functions u and v with respect to x. We have:
du/dx = 1
dv/dx = -1/2 * (x^2 - 1)^(-3/2) * 2x
Step 3: Apply the chain rule formula
Apply the chain rule formula, which states that the derivative of y with respect to x (dy/dx) is equal to dy/du multiplied by du/dx, plus dy/dv multiplied by dv/dx.
dy/dx = (dy/du)*(du/dx) + (dy/dv)*(dv/dx)
Step 4: Calculate the derivatives of y with respect to u and v
To find dy/du, differentiate y with respect to u while treating v as a constant:
dy/du = 1/v
To find dy/dv, differentiate y with respect to v while treating u as a constant:
dy/dv = -u/v^2
Step 5: Substitute the derivative values into the chain rule formula
Substitute the derivative values into the chain rule formula:
dy/dx = (1/v)*(du/dx) + (-u/v^2)*(dv/dx)
Step 6: Simplify the expression
Substitute the values of u, v, du/dx, and dv/dx:
dy/dx = (1 / ((x^2 - 1)^(-1/2)) )*(1) + ( -x / ((x^2 - 1)^(-1/2))^2 ) * (-1/2) * (x^2 - 1)^(-3/2) * 2x
Simplifying further, we get:
dy/dx = (1 / ((x^2 - 1)^(-1/2)) ) - x^2 / (x^2 - 1)
Regarding your question about online tutoring, there are several platforms that offer free online tutoring through chatrooms. Some popular ones include Khan Academy, Chegg, and StudyGate. You can simply search for "free online tutoring chatroom" to find various options available. Additionally, many universities and colleges also provide online tutoring services for their students.