If you have studied math in school, then you have no doubt learned the rule of power to determine the derivative of simple functions. However, when the function contains a square root or square root sign, such as ${ displaystyle { sqrt {x}}}$Review the power rule for derivatives. The first rule you have probably learned for finding derivatives is the power rule. This line says that for a variable ${ displaystyle x}$Rewrite the square root as an exponent. To find the derivative of a square root function, remember that the square root of a number or variable can also be written as an exponent. The term under the root sign is written as base, raised to the power of 1/2. The term is also used as an exponent of the square root. Take a look at the following examples:

• ${ displaystyle { sqrt {x}} = x ^ { frac {1} {2}}}$Apply the rule of power. If the function is the simplest square root, ${ displaystyle f (x) = { sqrt {x}}}$Simplify the result. At this stage, you should know that a negative exponent means taking the inverse of what the number would be with the positive exponent. The exponent of ${ displaystyle - { frac {1} {2}}}$Review the chain rule for features. The chain rule is a rule for derivatives that you use when the original function combines a function within another function. The chain rule says that, for two functions ${ displaystyle f (x)}$Define the functions for the chain rule. Using the chain rule requires that you first define the two functions that make up your combined function. For square root functions, the outer function is ${ displaystyle f (g)}$Determines the derivatives of the two functions. To apply the chain rule to the square root of a function, you must first find the derivative of the general square root function:
  • ${ displaystyle f (g) = { sqrt {g}} = g ^ { frac {1} {2}}}$Combine the functions in the chain rule. The chain rule is ${ displaystyle y ^ { prime} = f ^ { prime} (g) * g ^ { prime} (x)}$Determine derivatives of a root function using a quick method. When you want to find the derivative of the square root of a variable or a function, you can apply a simple rule: the derivative will always be the derivative of the number below the square root, divided by double the original square root. Symbolically, this can be represented as:
    • If ${ displaystyle f (x) = { sqrt {u}}}$Find the derivative of the number under the square root sign. This is a number or function under the square root sign. To use this quick method, find only the derivative of the number below the square root sign. Consider the following examples:
      • In the position ${ displaystyle { sqrt {5x + 2}}}$Write the derivative of the square root number as the numerator of a fraction. The derivative of a root function will contain a fraction. The numerator of this fraction is the derivative of the square root number. So, in the example functions above, the first part of the derivative will go like this:
        • If ${ displaystyle f (x) = { sqrt {5x + 2}}}$Write the denominator as double the original square root. With this quick method, the denominator is twice the original square root function. So, in the three example functions above, the denominators of the derivatives are:
          • If ${ displaystyle f (x) = { sqrt {5x + 2}}}$Combine the numerator and denominator to find the derivative. Put the two halves of the fraction together and the result will be the derivative of the original function.
            • If ${ displaystyle f (x) = { sqrt {5x + 2}}}$, than ${ displaystyle f ^ { prime} (x) = { frac {5} {2 { sqrt {5x + 2}}}}}$
            • If ${ displaystyle f (x) = { sqrt {3x ^ {4}}}}$, than ${ displaystyle f ^ { prime} (x) = { frac {12x ^ {3}} {2 { sqrt {3x ^ {4}}}}}}$
            • If ${ displaystyle f (x) = { sqrt { sin (x)}}}$, than ${ displaystyle f ^ { prime} (x) = { frac { cos (x)} {2 { sqrt { sin (x)}}}}}$