You are asked to derive expressions for the average rates of...

You are asked to derive expressions for the average rates of change of the functions $\sin x, \cos x,$ and $\tan x$ In each case, assume that the interval is $[x, x+h] .$ (The results are used in calculus in the study of derivatives.) Let $T(x)=\tan x .$ Show that $$ \frac{\Delta T}{\Delta x}=\frac{\tan h}{h} \cdot \frac{\sec ^{2} x}{1-\tan h \tan x} $$

You are asked to derive expressions for the average rates of change of the functions $\sin x, \cos x,$ and $\tan x$
In each case, assume that the interval is $[x, x+h] .$ (The results are used in calculus in the study of derivatives.)
Let $T(x)=\tan x .$ Show that
$$
\frac{\Delta T}{\Delta x}=\frac{\tan h}{h} \cdot \frac{\sec ^{2} x}{1-\tan h \tan x}
$$

Key Concepts

Secant Squared Function

The secant squared function, sec^2 x, appears in the derivative of the tangent function. Recognizing that the derivative of tan x is sec^2 x is important in calculus, and this knowledge aids in simplifying expressions obtained from the difference quotient form of tan x.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values in their domains. They are used extensively in calculus to simplify expressions and manipulate equations, especially when dealing with periodic functions like sine, cosine, and tangent.

Tangent Addition Formula

The tangent addition formula, which states that tan(x+h) = (tan x + tan h)/(1 - tan x tan h), is a trigonometric identity that enables the rewriting of expressions involving tan(x+h). This formula is critical for deriving the expression for the average rate of change of the tangent function over an interval.

Average Rate of Change

The average rate of change of a function over an interval measures how much the function value changes per unit change in the variable. In the context of calculus, it is seen as the slope of the secant line between two points on the graph of the function.

Difference Quotient

The difference quotient is the expression (f(x+h) - f(x))/h used to capture the average rate of change of a function over an interval. It is a central concept in calculus, serving as the foundational idea behind the derivative when we consider the limit as h approaches 0.

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