The multiple definitions of derivatives include, a word derived from another or from a root in the same or another language, something that is based on another source, and an arrangement or instrument (such as a future, option, or warrant) whose value derives from and is dependent on the value of an underlying asset. ! The concept of derivative used in Calculus is the stepping stone to typing all different modern mathematics together. Derivatives are the heart of Calculus, and the definition of a derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). ! Simply, a derivative is the limit of the average rate of change in the function as the length of the interval gets closer and closer to zero.
You can solve derivatives in 3 different ways:
1. Numerically
To find instantaneous ROC numerically, it is necessary to find a point (use h as variable) on the function that is as close to the point you are trying to solve the derivative of, in this case, always 0. The limit (h value) will be approaching 0. You should do this process three times with three different h’s getting closer to 0 to get a good approximate average.
This directly solves for a value that is close to the instantaneous ROC at the point. To do that we will use values that get infinitely closer to 0 and use the slope formula. For example:
Find f’(2) when f(x)=4x^2+2 numerically: *Recall: Slope formula = (y2-y1)/(x2-x1) This process can be summed up in a limit equation:
1. Numerically
To find instantaneous ROC numerically, it is necessary to find a point (use h as variable) on the function that is as close to the point you are trying to solve the derivative of, in this case, always 0. The limit (h value) will be approaching 0. You should do this process three times with three different h’s getting closer to 0 to get a good approximate average.
This directly solves for a value that is close to the instantaneous ROC at the point. To do that we will use values that get infinitely closer to 0 and use the slope formula. For example:
Find f’(2) when f(x)=4x^2+2 numerically: *Recall: Slope formula = (y2-y1)/(x2-x1) This process can be summed up in a limit equation:
2. Algebraically
Another way to solve derivatives is to manipulate it algebraically. This method is the most accurate to finding a close to exact solution. By solving algebraically you find the derivative equation. Each function has a derivative function that can tell us the derivative of any point on that function. Once you have the derivative function, you can find the instantaneous rate of change for any x value you plug in. To solve for a derivative equation algebraically, you must use this equation:
Another way to solve derivatives is to manipulate it algebraically. This method is the most accurate to finding a close to exact solution. By solving algebraically you find the derivative equation. Each function has a derivative function that can tell us the derivative of any point on that function. Once you have the derivative function, you can find the instantaneous rate of change for any x value you plug in. To solve for a derivative equation algebraically, you must use this equation:
3. The Power Rule
The algebraic process for finding the derivative is often time consuming due to the amount of work necessary in each step. The power rule states that to find the derivative function of a function, you can take the exponent of each value of x and multiply it by the coefficient of x at that part of the function, then subtract one from the exponent. A general form you can use for the power rule is this: f’(x)=n*a*x^n-1 , where ‘n’ represents the original power of the function, and ‘a’ represents the coefficient. (Note: Any constant that is not attached to an x, will always be 0) (Keep in mind that this rule doesn’t work for any function that is not a power function.) For example, find the derivative of the equation f(x)=23x^3+3+4x^5 using the power rule:
The algebraic process for finding the derivative is often time consuming due to the amount of work necessary in each step. The power rule states that to find the derivative function of a function, you can take the exponent of each value of x and multiply it by the coefficient of x at that part of the function, then subtract one from the exponent. A general form you can use for the power rule is this: f’(x)=n*a*x^n-1 , where ‘n’ represents the original power of the function, and ‘a’ represents the coefficient. (Note: Any constant that is not attached to an x, will always be 0) (Keep in mind that this rule doesn’t work for any function that is not a power function.) For example, find the derivative of the equation f(x)=23x^3+3+4x^5 using the power rule:
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