In calculus, the derivative of a function is a central concept that describes the rate of change of a function near a certain point. For trigonometric functions, especially tanx (tangent functions), understanding their derivatives is essential to understanding their properties and applications. So, what is the derivative of tanx? This article will provide an in-depth and introduction to this.
First, we need to clarify the definition of tanx. tanx is a tangent function and is defined as the tangent of any angle x equal to the ratio of its sine value to its cosine value, i.e., tanx = sinx cosx.
To find the derivative of tanx, we can use the formula for the derivative of the quotient, i.e., (u v).' = (u'v - uv'v 2, where you and v are derivables. In this case, let u = sinx and v = cosx.
First, we need to find the derivatives of you and v. According to the basic derivative formula, we know that the derivative of sinx is cosx, and the derivative of cosx is -sinx.
Substituting these values into the derivative formula of the quotient, we get:
tanx)' = (sinx)' * cosx - sinx * cosx)' / cos^2x
cosx * cosx - sinx * sinx) / cos^2x
cos^2x + sin^2x / cos^2x
Since the trigonometric identity sin 2x + cos 2x = 1, we can further simplify the above expression:
tanx)' = 1 / cos^2x
We know that cos 2x = 1 sec 2x, where secx is the secant function, i.e. the reciprocal of cosx. Therefore, we can write the above derivative expression as:
tanx)' = sec^2x
This is the derivative of tanx.
Summary:
By applying the derivative formula of the quotient and the basic derivative formula of trigonometric functions, we get the derivative of tanx as sec 2x. This result not only reveals the variation properties of the tanx function, but also has a wide range of applications in the fields of calculus, trigonometric identity transformation, and physics. Understanding and mastering the derivatives of tanx is of great significance for in-depth learning and application of trigonometric functions.