Find the derivatives of these:
f(x)=√
To find the derivative of a function, you can follow these general steps:
Step 1: Rewrite the function using exponent notation if necessary.
Step 2: Identify the function's outermost operation (e.g., square root, logarithm, etc.).
Step 3: Apply the corresponding derivative rule for that operation.
Step 4: Simplify the derivative if possible.
In the case of the square root function, f(x) = √x, here's how you can find its derivative:
Step 1: Rewrite the function using exponent notation: f(x) = x^(1/2)
Step 2: Identify the outermost operation, which is raising x to the power of 1/2.
Step 3: Apply the power rule for differentiation, which states that if f(x) = x^n, then f'(x) = nx^(n-1). In this case, n = 1/2, so applying the power rule gives f'(x) = (1/2) * x^(1/2 - 1) = (1/2) * x^(-1/2).
Step 4: Simplify the derivative: f'(x) = (1/2) * x^(-1/2) = (1/2) / √x = 1 / (2√x).
Therefore, the derivative of f(x) = √x is f'(x) = 1 / (2√x).