f(x)=xa{\displaystyle f(x)=x^{a}} f′(x)=axa−1{\displaystyle f^{\prime }(x)=ax^{a-1}} For example, review the following functions and their derivatives: If f(x)=x2{\displaystyle f(x)=x^{2}}, then f′(x)=2x{\displaystyle f^{\prime }(x)=2x} If f(x)=3x2{\displaystyle f(x)=3x^{2}}, then f′(x)=2∗3x=6x{\displaystyle f^{\prime }(x)=23x=6x} If f(x)=x3{\displaystyle f(x)=x^{3}}, then f′(x)=3x2{\displaystyle f^{\prime }(x)=3x^{2}} If f(x)=12x4{\displaystyle f(x)={\frac {1}{2}}x^{4}}, then f′(x)=4∗12x3=2x3{\displaystyle f^{\prime }(x)=4{\frac {1}{2}}x^{3}=2x^{3}}

x=x12{\displaystyle {\sqrt {x}}=x^{\frac {1}{2}}} 4=412{\displaystyle {\sqrt {4}}=4^{\frac {1}{2}}} 3x=(3x)12{\displaystyle {\sqrt {3x}}=(3x)^{\frac {1}{2}}}

f(x)=x     {\displaystyle f(x)={\sqrt {x}}\ \ \ \ \ }(Write the original function. ) f(x)=x(12)     {\displaystyle f(x)=x^{({\frac {1}{2}})}\ \ \ \ \ }(Rewrite the radical as an exponent. ) f′(x)=12x(12−1)   {\displaystyle f^{\prime }(x)={\frac {1}{2}}x^{({\frac {1}{2}}-1)}\ \ \ }(Find derivative with the power rule. ) f′(x)=12x(−12)   {\displaystyle f^{\prime }(x)={\frac {1}{2}}x^{(-{\frac {1}{2}})}\ \ \ }(Simplify exponent. )

Continuing with the square root of x function from above, the derivative can be simplified as: f′(x)=12x−12{\displaystyle f^{\prime }(x)={\frac {1}{2}}x^{-{\frac {1}{2}}}} f′(x)=12∗1x{\displaystyle f^{\prime }(x)={\frac {1}{2}}*{\frac {1}{\sqrt {x}}}} f′(x)=12x{\displaystyle f^{\prime }(x)={\frac {1}{2{\sqrt {x}}}}}

If y=f(g(x)){\displaystyle y=f(g(x))}, then y′=f′(g)∗g′(x){\displaystyle y^{\prime }=f^{\prime }(g)*g^{\prime }(x)}.

For example, suppose you wish to find the derivative of 3x+2{\displaystyle {\sqrt {3x+2}}}. Define the two parts as follows: f(g)=g=g12{\displaystyle f(g)={\sqrt {g}}=g^{\frac {1}{2}}} g(x)=(3x+2){\displaystyle g(x)=(3x+2)}

f(g)=g=g12{\displaystyle f(g)={\sqrt {g}}=g^{\frac {1}{2}}} f′(g)=12g−12{\displaystyle f^{\prime }(g)={\frac {1}{2}}g^{-{\frac {1}{2}}}} f′(g)=12g{\displaystyle f^{\prime }(g)={\frac {1}{2{\sqrt {g}}}}} Then find the derivative of the second function: g(x)=(3x+2){\displaystyle g(x)=(3x+2)} g′(x)=3{\displaystyle g^{\prime }(x)=3}

y′=12g∗3{\displaystyle y^{\prime }={\frac {1}{2{\sqrt {g}}}}*3} y′=12(3x+2∗3{\displaystyle y^{\prime }={\frac {1}{2{\sqrt {(3x+2}}}}*3} y′=32(3x+2{\displaystyle y^{\prime }={\frac {3}{2{\sqrt {(3x+2}}}}}

If f(x)=u{\displaystyle f(x)={\sqrt {u}}}, then f′(x)=u′2u{\displaystyle f^{\prime }(x)={\frac {u^{\prime }}{2{\sqrt {u}}}}}

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)}.

If f(x)=5x+2{\displaystyle f(x)={\sqrt {5x+2}}}, then f′(x)=5denom{\displaystyle f^{\prime }(x)={\frac {5}{\text{denom}}}} If f(x)=3x4{\displaystyle f(x)={\sqrt {3x^{4}}}}, then f′(x)=12x3denom{\displaystyle f^{\prime }(x)={\frac {12x^{3}}{\text{denom}}}} If f(x)=sin⁡(x){\displaystyle f(x)={\sqrt {\sin(x)}}}, then f′(x)=cos⁡(x)denom{\displaystyle f^{\prime }(x)={\frac {\cos(x)}{\text{denom}}}}

For f(x)=5x+2{\displaystyle f(x)={\sqrt {5x+2}}}, then f′(x)=num25x+2{\displaystyle f^{\prime }(x)={\frac {\text{num}}{2{\sqrt {5x+2}}}}} If f(x)=3x4{\displaystyle f(x)={\sqrt {3x^{4}}}}, then f′(x)=num23x4{\displaystyle f^{\prime }(x)={\frac {\text{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)={\frac {\text{num}}{2{\sqrt {\sin(x)}}}}}

For f(x)=5x+2{\displaystyle f(x)={\sqrt {5x+2}}}, then f′(x)=525x+2{\displaystyle f^{\prime }(x)={\frac {5}{2{\sqrt {5x+2}}}}} If f(x)=3x4{\displaystyle f(x)={\sqrt {3x^{4}}}}, then f′(x)=12x323x4{\displaystyle f^{\prime }(x)={\frac {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)={\frac {\cos(x)}{2{\sqrt {\sin(x)}}}}}