6 Trig Derivatives Of Trigonometric Functions, Derivatives Of Trigonometric Functions

Derivative of trig functions

Before we start learning how to take derivative of trig functions, why don't we go back to the basics? Going back and reviewing the basics is always a good thing. This is because a lot of people tend to forget about the properties of trigonometric functions. In addition, forgetting certain trig properties, identities, and trig rules would make certain questions in Calculus even more difficult to solve. Let's first take a look at the six trigonometric functions.

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The 6 Trigonometric Functions

The first trigonometric function we will be looking at is f(x)=sin⁡xf(x) = sin xf(x)=sinx. If we are to graph the function, we will get this:

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Graph 1: sinx

Notice that the function is continuous from <-∞,∞infty, infty∞,∞>, and it's a nice smooth curve with no sharp turns. This means that sin⁡xsin xsinx is differentiable at every point, and so we will not have to worry about getting something undefined. In addition, we know that the slope at x=π2±πn,n∈Ix = frac{pi}{2} pm pi n, n in Ix=2π​±πn,n∈I are 000. Hence, the derivative of sin⁡xsin xsinx will always be zero at those points.

Next is the function f(x)=cos⁡xf(x) = cos xf(x)=cosx. If we were graph the function, we will get:

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Graph 2: cosx

Again note that the function is continuous from <-∞,∞infty, infty∞,∞>, and it has a nice smooth curve. So cos⁡xcos xcosx is also differentiable at every point. In addition, the slope is equal to 0 at xxx = 0 ±πn,n∈Ipm pi n, n in I±πn,n∈I. Hence, the derivative will of cos⁡xcos xcosx will be 0 at those points.

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Now the function y=tan⁡xy = an xy=tanx is a bit more interesting. It can be rewritten as y=sin⁡xcos⁡xy= frac{sin x}{cos x}y=cosxsinx​, the graph of this function looks like this:

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Graph 3: tanx

Notice the xxx values at x=π2±πn,n∈Ix = frac{pi}{2} pm pi n, n in Ix=2π​±πn,n∈I are undefined, and have vertical asymptotes. This means the derivative of tan⁡x an xtanx will be not differentiable at those points. One interesting thing to note here is that the slope of tan⁡x an xtanx is never negative or 0. Hence, we should expect the derivative of tan⁡x an xtanx to always be positive.

Next are the reciprocal functions of sin cos tan. First, the reciprocal of sin⁡xsin xsinx is csc⁡xcsc xcscx. We see the graph of f(x)=csc⁡xf(x) = csc xf(x)=cscx looks like this:

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Graph 4: cscx

Again, there are vertical asymptotes at xxx = 0 ±πn,n∈Ipm pi n, n in I±πn,n∈I. So they are not differentiable at those points. In addition, the slope of the function at x=π2±πn,n∈Ix = frac{pi}{2} pm pi n, n in Ix=2π​±πn,n∈I are 0. So the derivatives are always 0 at those points.

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Now the next function is f(x)=sec⁡xf(x) = sec xf(x)=secx, which is the reciprocal of cos⁡xcos xcosx. Graphing gives:

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Graph 5: secx

This is very similar to csc⁡xcsc xcscx, but the difference is that the role's of the xxx values has been switched. Now the xxx values at x=π2±πn,n∈Ix = frac{pi}{2} pm pi n, n in Ix=2π​±πn,n∈I are the vertical asymptotes and the xxx value at x=0±πn,n∈Ix = 0 pm pi n, n in Ix=0±πn,n∈I are when the tangent slopes of the function are 0. Hence the derivative of the function is not differentiable at x=π2±πn,n∈Ix = frac{pi}{2} pm pi n, n in Ix=2π​±πn,n∈I and the derivative is 0 at x=0±πn,n∈Ix = 0 pm pi n, n in Ix=0±πn,n∈I.

The last function is y=cot⁡xy = cot xy=cotx.

Note that the vertical asymptotes are at x=0±πn,n∈Ix = 0 pm pi n, n in Ix=0±πn,n∈I and the slope of this never positive or 0. Hence, the derivative of this function is always positive, and not differentiable at x=0±πn,n∈Ix = 0 pm pi n, n in Ix=0±πn,n∈I.

Now that we are finished looking at the 6 trig functions, let's now review some of the trigonometric identities that come in handy when taking derivatives.

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Trigonometric identities

The first six identities are reciprocal identities, which come in handy when you want your derivatives in a certain form.

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