Article: Whenever you wish to find the derivative of the square root of a variable or a function, you can apply a simple pattern. The derivative will always be the derivative of the radicand, divided by double the original square root. Symbolically, this can be shown as: If f(x)=u{\displaystyle f(x)={\sqrt {u}}}, then f(x)=u2u{\displaystyle f^{\prime }(x)={
rac {u^{\prime }}{2{\sqrt {u}}}}} The radicand is the term or function underneath the square root sign. To apply this shortcut, find the derivative of the radicand alone. Consider the following examples: In the function 5x+2{\displaystyle {\sqrt {5x+2}}}, the radicand is (5x+2){\displaystyle (5x+2)}. Its derivative is 5{\displaystyle 5}. In the function 3x4{\displaystyle {\sqrt {3x^{4}}}}, the radicand is 3x4{\displaystyle 3x^{4}}. Its derivative is 12x3{\displaystyle 12x^{3}}. In the function sin(x){\displaystyle {\sqrt {sin(x)}}}, the radicand is sin(x){\displaystyle \sin(x)}. Its derivative is cos(x){\displaystyle \cos(x)}. The derivative of a radical function will involve a fraction. The numerator of this fraction is the derivative of the radicand. Thus, for the sample functions above, the first part of the derivative will be as follows: If f(x)=5x+2{\displaystyle f(x)={\sqrt {5x+2}}}, then f(x)=5denom{\displaystyle f^{\prime }(x)={
rac {5}{ ext{denom}}}} If f(x)=3x4{\displaystyle f(x)={\sqrt {3x^{4}}}}, then f(x)=12x3denom{\displaystyle f^{\prime }(x)={
rac {12x^{3}}{ ext{denom}}}} If f(x)=sin(x){\displaystyle f(x)={\sqrt {\sin(x)}}}, then f(x)=cos(x)denom{\displaystyle f^{\prime }(x)={
rac {\cos(x)}{ ext{denom}}}} Using this shortcut, the denominator will be two times the original square root function. Thus, for the three sample functions above, the denominators of the derivatives will be: For f(x)=5x+2{\displaystyle f(x)={\sqrt {5x+2}}}, then f(x)=num25x+2{\displaystyle f^{\prime }(x)={
rac { ext{num}}{2{\sqrt {5x+2}}}}} If f(x)=3x4{\displaystyle f(x)={\sqrt {3x^{4}}}}, then f(x)=num23x4{\displaystyle f^{\prime }(x)={
rac { ext{num}}{2{\sqrt {3x^{4}}}}}} If f(x)=sin(x){\displaystyle f(x)={\sqrt {\sin(x)}}}, then f(x)=num2sin(x){\displaystyle f^{\prime }(x)={
rac { ext{num}}{2{\sqrt {\sin(x)}}}}} Put the two halves of the fraction together, and the result will be the derivative of the original function. For f(x)=5x+2{\displaystyle f(x)={\sqrt {5x+2}}}, then f(x)=525x+2{\displaystyle f^{\prime }(x)={
rac {5}{2{\sqrt {5x+2}}}}} If f(x)=3x4{\displaystyle f(x)={\sqrt {3x^{4}}}}, then f(x)=12x323x4{\displaystyle f^{\prime }(x)={
rac {12x^{3}}{2{\sqrt {3x^{4}}}}}} If f(x)=sin(x){\displaystyle f(x)={\sqrt {\sin(x)}}}, then f(x)=cos(x)2sin(x){\displaystyle f^{\prime }(x)={
rac {\cos(x)}{2{\sqrt {\sin(x)}}}}}

What is a summary?
Learn the shortcut for derivatives of any radical function. Find the derivative of the radicand. Write the derivative of the radicand as the numerator of a fraction. Write the denominator as double the original square root. Combine numerator and denominator to find the derivative.