How do i find the second derivative of f(x)= tan(1-x)^1/2
f'(x) = (1/2)(tan(1-x))^(-1/2)*sec^2 (1-x)(-1)
= -(1/2) [tan(1-x)]^(-1/2) [sec(1-x)]^2
now we need the product rule, what a mess ...
f''(x) = -(1/2) [tan(1-x)]^(-1/2) (2)(sec(1-x)) (sec(1-x))(tan(1-x))(-1) + (-1/2)[sec(1-x)]^2 (-1/2)tan(1-x))^(-3/2)(sec^2(1-x))(-1)
check this carefully and see if you simplify by factoring. I can see all kinds of common factors, but find it hard to read my own typing, lol
To find the second derivative of a function, you need to follow a specific procedure. In this case, we want to find the second derivative of the function f(x) = tan((1-x)^(1/2)).
Step 1: Start by finding the first derivative of f(x) using the chain rule. The chain rule states that if we have a composition of two functions, say f(g(x)), then the derivative of f with respect to x is equal to the derivative of f with respect to g multiplied by the derivative of g with respect to x.
In our case, let's define u = (1-x)^(1/2). Applying the chain rule, we have:
f'(x) = sec^2(u) * (1/2)(1-x)^(-1/2) * (-1) = -(1/2)sec^2(u)(1-x)^(-1/2).
Step 2: Next, we find the second derivative by differentiating f'(x) with respect to x. Again, using the chain rule, we get:
f''(x) = d/dx(-(1/2)sec^2(u)(1-x)^(-1/2)).
= -(1/2) * [sec^2(u) * d/dx((1-x)^(-1/2))].
Step 3: Now we need to find d/dx((1-x)^(-1/2)). To do this, apply the chain rule again, considering v = 1 - x:
d/dx((1-x)^(-1/2)) = d/du(u^(-1/2)) * d/dx(1 - x).
The first part of the expression, d/du(u^(-1/2)), is more straightforward. It is equal to (-1/2)u^(-3/2).
The second part, d/dx(1 - x) = -1, as the derivative of a constant is zero.
Step 4: Substitute the values we found in the previous steps back into the expression we obtained in Step 2 for the second derivative:
f''(x) = -(1/2) * [sec^2(u) * (-1/2)u^(-3/2) * (-1)].
= (1/4) * sec^2(u)u^(-3/2).
Step 5: Finally, rewrite the expression in terms of x:
f''(x) = (1/4) * sec^2((1-x)^(1/2)) * ((1-x)^(1/2))^(-3/2).
That's it! Now you have the second derivative of the function f(x)= tan((1-x)^(1/2)).