Formal definition of derivative: trig vs non trig functions

In summary, the formal definition of a derivative, which is formulated from the cartesian coordinate system, can also be applied to trigonometric functions. Although trig functions are represented on a graph with angles, the argument to these functions is actually dimensionless, making them similar to ordinary real numbers. This allows them to be set up into the formal definition of a derivative in the same manner as non-trig functions. While this may not seem as intuitive visually, both types of functions can be thought of as essentially the same thing and can be represented in the formal definition formula.
  • #1
LearninDaMath
295
0
for derivative sinx = cosx, by setting up into formal definition formula limΔx->0 [itex]\frac{f(x+Δx)-f(x)}{Δx}[/itex]


this formal definition of derivative is formulated from the cartesian coordinate system where the horizontal is x and verticle is y. But sinx is a trig function and trig functions are represented on the graph where the horizontal is an angle.

So how does it make sense that trig functions can be set up into the formal definition of a derivative in the same manner as non-trig functions?

Meaning, how does this make sense? limΔx->0 [itex]\frac{sin(x+Δx)-sinx}{Δx}[/itex]

I know how to work it out and get cosx. But it doesn't seem to make as much sense visually as a nontrig proof.
 
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  • #2
LearninDaMath said:
for derivative sinx = cosx, by setting up into formal definition formula limΔx->0 [itex]\frac{f(x+Δx)-f(x)}{Δx}[/itex]


this formal definition of derivative is formulated from the cartesian coordinate system where the horizontal is x and verticle is y. But sinx is a trig function and trig functions are represented on the graph where the horizontal is an angle.
Not really. The argument to the standard trig functions is radians, which are dimensionless, which makes them just like ordinary real numbers.
LearninDaMath said:
So how does it make sense that trig functions can be set up into the formal definition of a derivative in the same manner as non-trig functions?

Meaning, how does this make sense? [itex]\frac{sin(x+Δx)-sinx}{Δx}[/itex]

I know how to work it out and get cosx. But it doesn't seem to make as much sense visually as a nontrig proof.
 
  • #3
Thanks Mark44, when thinking about it, it seems to be the same thing after all. The horizontal is still x in either case like you say. And since a trig function is just a function, sin(x) can be thought of as f(x). So both types of functions are essentially the same thing, such as in terms of representing two points on each respective function (trig or non-trig) in the formal defintion formula. Thanks, Mark, appreciate it.
 

1. What is the formal definition of derivative for trigonometric functions?

The formal definition of derivative for trigonometric functions is the limit of the change in the function divided by the change in the input value, as the change in the input value approaches zero. In other words, it is the slope of the tangent line to the curve at a specific point.

2. How does the formal definition of derivative differ for non-trigonometric functions?

The formal definition of derivative for non-trigonometric functions is the same as for trigonometric functions, but it may involve more complex calculations depending on the specific function. In general, it involves finding the limit of the difference quotient as the change in the input value approaches zero.

3. Why is the formal definition of derivative important?

The formal definition of derivative is important because it is the foundation of calculus and is used to calculate rates of change, slopes of curves, and optimization problems. It also allows us to understand the behavior of functions and make predictions about their values.

4. Can the formal definition of derivative be applied to all functions?

Yes, the formal definition of derivative can be applied to all functions, both trigonometric and non-trigonometric. However, the process of calculating the derivative may vary depending on the type of function and may require more complex techniques for some functions.

5. Are there any common misconceptions about the formal definition of derivative?

One common misconception about the formal definition of derivative is that it is only used to find the slope of a curve at a specific point. In reality, it can also be used to find the slope of a tangent line at any point on the curve, as well as the overall rate of change of the function. Additionally, some people may think that the formal definition of derivative only applies to continuous functions, but it can also be applied to discontinuous functions by using the limit definition.

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