sinx 求导为什么是 cosx(求sinx的导数为什么是cosx？)

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IntroductionWhenstudyingcalculus,oneofthefundamentalconceptsisthederivative,whichmeasurestherateofchangeofafunction.Oneofthemostbasicandessentialderiv

Introduction

When studying calculus, one of the fundamental concepts is the derivative, which measures the rate of change of a function. One of the most basic and essential derivatives is the derivative of the sine function, which yields the cosine function. In this article, we will explore why the derivative of sinx is cosx and how we can obtain it using various math concepts.

Definition of Derivative

Before we dive into the derivative of sinx, let us revisit the definition of the derivative. The derivative of a function f(x) is denoted as f'(x), and it measures the rate of change of f(x) with respect to x. In other words, it tells us how fast or slow f(x) is changing at any given point x. Mathematically, we can represent the derivative of f(x) as:

f'(x) = lim(h->0) [(f(x+h) - f(x))/h]

This formula tells us that we need to take the limit as h approaches 0 of the ratio of the change in f(x) over a small change in x. The closer h gets to 0, the more accurate our measurement of the instantaneous rate of change of f(x) will be.

Derivative of Sine Function

Now, let us focus on the derivative of sinx. We can obtain it using the definition of the derivative that we just discussed. Let f(x) = sinx. Then:

f'(x) = lim(h->0) [(sin(x+h) - sinx)/h]

Using the angle sum formula for sine, we can rewrite the numerator as:

sin(x+h) - sinx = (sinx * cosh) + (cosx * sinh) - sinx = cosx * sinh

where cosh = cos(h) and sinh = sin(h). Therefore, f'(x) becomes:

f'(x) = lim(h->0) [cosx * (sinh/h)]

We can evaluate this limit using L'Hopital's rule, which states that if we have a limit of the form 0/0 or infinity/infinity, we can take the derivative of the numerator and denominator separately and evaluate the limit again. Applying L'Hopital's rule to f'(x), we get:

f'(x) = lim(h->0) [cosx * (sinh/h)] = lim(h->0) [cosx * cosh] = cosx * 1 = cosx

Therefore, we have shown that the derivative of sinx is equal to cosx using the definition of the derivative and some trigonometric identities.

Graphical Interpretation

We can also understand why the derivative of sinx is cosx from a graphical perspective. The sine function is a periodic function that oscillates between -1 and 1 as x varies. Its derivative, the cosine function, represents the slope of the tangent lines to the sine curve at each point x. For example, at x = 0, the slope of the tangent line to sinx is 1, which corresponds to the value of cosx. As x increases, the slope of the tangent line decreases, and the value of cosx decreases accordingly.

Furthermore, we can also see that the sine and cosine curves are orthogonal, meaning that they intersect at right angles. This fact is a consequence of the Pythagorean identity for trigonometric functions, which states that sin^2(x) + cos^2(x) = 1 for all values of x. Therefore, the derivative of sinx is cosx because the sine and cosine functions are closely related, with the cosine function representing the slope of the tangent lines to the sine curve.

Conclusion

In conclusion, we have explored why the derivative of sinx is cosx using various math concepts, such as the definition of the derivative, trigonometric identities, L'Hopital's rule, and graphical interpretations. The derivative of sinx is a fundamental concept in calculus that has broad applications in various fields of science and engineering. Understanding the relationship between the sine and cosine functions is essential for mastering calculus and building a solid foundation for advanced math concepts.

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