How to Differentiate Square Root Functions


In calculus, derivatives measure the rate of change in a function with respect to one of its variable, and the method you use to calculate derivatives is called differentiation. Differentiating a square root function is more complicated than a common function such as a quadratic because it acts as a function within a function. A square root function is a function raised to the power of half. Like in any other power function, we use the chain rule to differentiate a square root function.

  • Write the square root function. Suppose it is given by y = sqrt(x^5 + 3x -7).

  • Substitute the "inner expression" of x^5 + 3x -- 7 by "u." The function simplifies to become y = sqrt (u). Remember that square root is power raised to half. Therefore, it can be written as y = u^1/2.

  • Use chain rule to expand the function. The chain rule is given by dy/dx = dy/du du/dx. Applying it to the function in Step 2 gives dy/dx = [du^(1/2)/du] du/dx.

  • Differentiate the function with respect to "u." Here, dy/dx = 1/2 u^(1-1/2) du/dx. Simplify it further into dy/dx = 1/2 1/sqrt(u) du/dx.

  • Substitute back the "inner expression" from Step 2 in place of "u." Thus, dy/dx = 1/2 1/sqrt(x^5 + 3x -7) d(x^5 +3x -7)/dx.

  • Complete the remaining differentiation with respect to x to get the final answer. In this example, the derivative is given by dy/dx = 1/2 1/sqrt(x^5 + 3x -7) (5x +3).


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