Lesson 2.2: How to Differentiate: Monomials, Sine, and Cosine

We’ve just shown that the derivative of x^2 is 2x; symbolically, d/dx x^2=2x. Now, let’s expand our known derivatives to monomial functions. Recall that we define the derivative df/dxof a function f(x) as df/dx = lim(x->a)(f(x)-f(a))/(x-a) or df/dx = lim(h->0)(f(x+h)-f(x))/h.

Let’s begin with the most basic case of a monomial: one of the form f(x)=c for some constant c. If we graph y=f(x), the graph takes the shape of a straight horizontal line, which clearly has slope 0. Therefore d/dx c=0.

Moving on to the next-lowest-degree monomial: f(x)=cx. The graph of y=f(x) is a straight line with slope c at every point, so d/dx cx=c.

We already showed that d/dx x^2=2x, but now let’s try to expand our definition to find d/dx cx^2. So we want to evaluate lim(x->a)(cx^2-ca^2)/(x-a), or, factoring out c, lim(x->a)c(x^2-a^2)/(x-a). In fact, since c is unaffected as x approaches a, we can take it out of the limit altogether: c lim(x->a)(x^2-a^2)/(x-a). We’ve previously shown that lim(x->a)(x^2-a^2)/(x-a)=2x, so d/dx cx^2=2cx.

Now let’s try to find d/dx cx. We just demonstrated that you can take the c factor out of the limit when evaluating it, so we will just consider lim(x->a)(x^3-a^3)/(x-a). We factor the numerator: x^3-a^x=(x-a)(x^2+ax+a^x), making our limit lim(x->a)(x-a)(x^2+ax+a^2)/(x-a). The x-a factor cancels leaving x^a+ax+a^2, at which point we plug in a=x to give x^2+x^2+x^2=3x^2, so d/dx x^3=3x^2 and, in general, d/dx cx^3=3cx^2.

The process for evaluating d/dx cx^4 is similar: d/dx cx^4=c lim(x->a)(x^4-a^4)/(x-a); we factor the numerator to give c lim(x->a)(x-a)(x+a)(x^2+a^2)/(x-a); x-a cancels; the limit evaluates to d/dx cx^4=4cx^3.

Perhaps you’ve deduced a pattern already. In case you haven’t, here’s how the evaluation of d/dx cx^5 goes: d/dx cx^5=c lim(x->a)(x^5-a^5)/(x-a). Factor: x^5-a^5=(x-a)(x^4+ax^3+a^2x^2+a^3x+a^4). Cancel x-a. Plug in a=x and we have d/dx cx^5=5cx^4.

Let’s summarize our results so far:

f(x)f'(x)
c0
cxc
cx^22cx
cx^33cx^2
cx^44cx^3
cx^55cx^4

For the sake of simplicity and to make it easier to spot the pattern, let’s reexamine that table with the c removed and with a few rows rewritten slightly:

f(x)f'(x)
c0
x1x^0
x^22x^1
x^33x^2
x^44x^3
x^55x^4

Hopefully you see the pattern by now. To write it out mathematically, d/dx x^n=nx^(n-1). If we throw in a constant, d/dx cx^n=ncx^(n-1).

Now let’s turn our attention to the two basic trigonometric functions f(x)=sin x and g(x)=cos x. Recall that in calculus, we will work exclusively in radians. If we graph y=f(x) and plot the (estimated) slope of tangent lines at certain points, we might estimate that f'(0)=1, f'(pi/2)=0, f'(pi)=-1, f'(3pi/2)=0, and that the derivatives at points then cycle similarly: f'(2pi)=1, f'(5pi/2)=0, etc. Therefore the most basic conclusion we can draw is that f'(x) is also a periodic function.

Let’s work with the definition of a limit, df/dx=lim(h->0)(f(x+h)-f(x))/h. Plugging in for f(x)=sin x, d/dx sin x = lim(h->0)(sin(x+h)-sin x)/h. An identity tells us that sin(x+h)=sin h cos x + sin x cos h, so the limit becomes lim(h->0)(sin h cos x + sin x cos h - sin x)/h. We factor out a term of sin x in the numerator, giving lim(h->0)(sin h cos x + sin x (cos h - 1))/h. Now we split up the fraction and apply the sum law, and we have lim(h->0)(sin h cos x)/h + lim(h->0)(sin x (cos h - 1)). In the first term we can remove cos x from the limit because it does not depend on h, and we can remove sin x from the second term for the same reason. Now the limit is cos x lim(h->0)(sin h)/h + sin x lim(h->0)(cos h - 1). We can evaluate the second term right away: sin x lim(h->0)(cos h - 1) = sin x (cos(0)-1)=0. The remaining term in our limit is a bit trickier to formally prove, but in the second example of Lesson 2, we suggested that lim(x->0)(sin x)/x=1. If you’re interested, a geometric argument as well as a proof can be found in the textbook. Thus our first term is (1)cos x or just d/dx sin x = cos x.

We can go through similar tedium to find that d/dx cos x = -sin x. In our next lesson, we will learn how to differentiate functions that are simpler functions combined by addition, subtraction, multiplication, and division. This will enable us to find derivatives for the remaining trigonometric functions as well as polynomials, rational functions, and more.

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