How to Take Derivatives

Опубликовал Admin
28-04-2021, 22:40
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The derivative is an operator that finds the instantaneous rate of change of a quantity, usually a slope. Derivatives can be used to obtain useful characteristics about a function, such as its extrema and roots. Finding the derivative from its definition can be tedious, but there are many techniques to bypass that and find derivatives more easily.

Preliminaries

  1. Understand the definition of the derivative. While this will almost never be used to actually take derivatives, an understanding of this concept is vital nonetheless.
    • Recall that the linear function is of the form y=mx+b.{\displaystyle y=mx+b.} To find the slope m{\displaystyle m} of this function, two points on the line are taken, and their coordinates are plugged into the relation m=y2−y1x2−x1.{\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.} Of course, this can only be used with linear graphs.
    • For nonlinear functions, the line will be curved, so taking the difference of two points can only give the average rate of change between them. The line that intersects these two points is called the secant line, with a slope m=f(x+Δx)−f(x)Δx,{\displaystyle m={\frac {f(x+\Delta x)-f(x)}{\Delta x}},} where Δx=x2−x1{\displaystyle \Delta x=x_{2}-x_{1}} is the change in x,{\displaystyle x,} and we have replaced y{\displaystyle y} with f(x).{\displaystyle f(x).} This is the same equation as the one before.
    • The concept of the derivatives comes in when we take the limit Δx→0.{\displaystyle \Delta x\to 0.} When this happens, the distance between the two points shrinks, and the secant line better approximates the rate of change of the function. When we do send the limit to 0, we end up with the instantaneous rate of change and obtain the slope of the tangent line to the curve (see animation above). Then, we end up with the definition of the derivative, where the prime symbol denotes the derivative of the function f.{\displaystyle f.}
      • f′(x)=limΔx→0f(x+Δx)−f(x)Δx{\displaystyle f^{\prime }(x)=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}
    • Finding the derivative from this definition stems from expanding the numerator, canceling, and then evaluating the limit, since immediately evaluating the limit will give a 0 in the denominator.
  2. Understand the derivative notation. There are two common notations for the derivative, though there are others.
    • Lagrange's Notation. In the previous step, we used this notation to denote the derivative of a function f(x){\displaystyle f(x)} by adding a prime symbol.
      • f′(x){\displaystyle f^{\prime }(x)}
      • This notation is pronounced "f{\displaystyle f} prime of x.{\displaystyle x.}" To form higher order derivatives, simply add another prime symbol. When derivatives of fourth or higher order are taken, the notation becomes f(4)(x),{\displaystyle f^{(4)}(x),} where this represents the fourth derivative.
    • Leibniz's Notation. This is the other commonly used notation, and we will use it in the rest of the article.
      • dfdx{\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} x}}}
      • (For shorter expressions, the function can be placed in the numerator.) This notation literally means "the derivative of f{\displaystyle f} with respect to x.{\displaystyle x.}" It may be helpful to think of it as ΔyΔx{\displaystyle {\frac {\Delta y}{\Delta x}}} for values of x{\displaystyle x} and y{\displaystyle y} that are infinitesimally different from each other. When using this notation for higher derivatives, you must write d2fdx2,{\displaystyle {\frac {\mathrm {d} ^{2}f}{\mathrm {d} x^{2}}},} where this represents the second derivative.
      • (Note that there "should" be parentheses in the denominator, but no one ever writes them, since everyone understands what we mean without them anyway.)

Basic Techniques

  1. Substitute (x+Δx){\displaystyle (x+\Delta x)} into the function. For this example, we will define f(x)=2x2+6x.{\displaystyle f(x)=2x^{2}+6x.}
    • f(x+Δx)=2(x+Δx)2+6(x+Δx)=2(x2+2xΔx+(Δx)2)+6x+6Δx=2x2+4xΔx+2(Δx)2+6x+6Δx.{\displaystyle {\begin{aligned}f(x+\Delta x)&=2(x+\Delta x)^{2}+6(x+\Delta x)\\&=2(x^{2}+2x\Delta x+(\Delta x)^{2})+6x+6\Delta x\\&=2x^{2}+4x\Delta x+2(\Delta x)^{2}+6x+6\Delta x.\end{aligned}}}
  2. Substitute the function into the limit. Then evaluate the limit.
    • ddxf(x)=limΔx→0(2x2+4xΔx+2(Δx)2+6x+6Δx)−(2x2+6x)Δx=limΔx→04xΔx+2(Δx)2+6ΔxΔx=limΔx→0Δx(4x+2Δx+6)Δx=limΔx→04x+2Δx+6=4x+6.{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}f(x)&=\lim _{\Delta x\to 0}{\frac {(2x^{2}+4x\Delta x+2(\Delta x)^{2}+6x+6\Delta x)-(2x^{2}+6x)}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {4x\Delta x+2(\Delta x)^{2}+6\Delta x}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {\Delta x(4x+2\Delta x+6)}{\Delta x}}\\&=\lim _{\Delta x\to 0}4x+2\Delta x+6\\&=4x+6.\end{aligned}}}
    • This is a lot of work for such a simple function. We will see that there are plenty of derivative rules to skirt past this type of evaluation.
    • You can find the slope anywhere on the function f(x)=2x2+6x.{\displaystyle f(x)=2x^{2}+6x.} Simply plug in any x value into the derivative df(x)dx=4x+6.{\displaystyle {\frac {\mathrm {d} f(x)}{\mathrm {d} x}}=4x+6.}

Using a Calculator

  1. Press Alpha F2. This will open the “Window” key, where you’ll see lots of options. Scroll over to the FUNC tab if you aren’t there already.
    • These instructions are for new models of the TI-84 and the TI-84 Plus. Older models may be slightly different.
  2. Select nDeriv(. It’s the third option on the list. When you get to it, you can press “enter” to select it.
  3. Enter your formula into the equation. When you hit the derivative option, your calculator will give you a blank equation that looks like this: (d/d[])([])|x=[]{\displaystyle (d/d[])([])|x=[]}. Go ahead and enter your specific numbers into the equation.
    • For example, if you were finding the derivative of the function x2{\displaystyle x^{2}} where x=2{\displaystyle x=2}, you’d enter (d/dx)(x2)|x=2{\displaystyle (d/dx)(x^{2})|x=2}.
    • If you have an equation plotted in the Y plots of your calculator, you can enter those into a blank field by pressing vars > Y-VARS > Function.
  4. Hit “enter” to find the derivative. Once you have all of your numbers entered, you can select “enter” on your calculator to get your answer. It will (hopefully) give you your answer in an easy to understand whole number.
    • For example, in the equation above, the derivative is 4.

Tips

  • Every technique outlined in this article on calculating derivatives can be verified by a proper use of the definition of the derivative. If, for example, the power rule seems sketchy to you, try and recover the formula using the definition.
  • Practice the product rule, chain rule, and especially implicit differentiation, as these are more difficult to differentiate and are widely used outside mathematics.

Warnings

  • Some students will be tempted to use programs on their calculators to take derivatives. While these programs are very useful for confirming your answers, you should not rely on these. Make sure you understand the concepts of deriving and are able to do it yourself.
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