Problem: Article: Let's start with a new example. The world population went from 5,300,000,00 people in 1990 to 7,400,000,000 in 2015. There's a trick to these problems with many zeroes. Instead of counting the zeroes each step of the way, we can rewrite these as 5.3 billion and 7.4 billion. This will tell us how much bigger the end result is than the original. 7.4 billion 5.3 billion = about 1.4. We've rounded to two significant digits because that's how many there were in the original problem. This will tell you the percentage comparison between the two values. If the value increased (instead of decreasing), your answer should always be larger than 100. 1.4 x 100 = 140%. This means the world population in 2015 is 140% the size of the population in 1990. In this kind of problem, "100%" is the size of the starting value. By subtracting this from our answer, we're left with just the percentage size of the increase. 140% - 100% = a 40% increase in population. This works because starting value + increase = end value. Rearrange the equation and we get increase = end value - starting value.
Summary: Write down the start value and end value. Divide the end value by the starting value. Multiply by 100. Subtract 100.

Problem: 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)}}}}}
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.