Question
find the derivative of Sqrt[1-Power[x,2]]
Answers
Answered by
GPT 3.5
To find the derivative of the function f(x) = √(1 - x^2), we can use the chain rule.
Let u = 1 - x^2. Taking the derivative of u with respect to x, we get du/dx = -2x.
So, by the chain rule, the derivative of f(x) with respect to x is given by:
df/dx = df/du * du/dx
df/du = 1/2 √u (by applying the power rule for differentiation of a square root)
Therefore,
df/dx = 1/2 √u * du/dx
= 1/2 √(1 - x^2) * (-2x)
= -x/√(1 - x^2)
Hence, the derivative of √(1 - x^2) is -x/√(1 - x^2).
Let u = 1 - x^2. Taking the derivative of u with respect to x, we get du/dx = -2x.
So, by the chain rule, the derivative of f(x) with respect to x is given by:
df/dx = df/du * du/dx
df/du = 1/2 √u (by applying the power rule for differentiation of a square root)
Therefore,
df/dx = 1/2 √u * du/dx
= 1/2 √(1 - x^2) * (-2x)
= -x/√(1 - x^2)
Hence, the derivative of √(1 - x^2) is -x/√(1 - x^2).