Glossary
Chain Rule
The Chain Rule is a formula used to find the derivative of a composite function. It states that the derivative of $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.
Example:
To differentiate , you must apply the \textbf{Chain Rule}, resulting in .
Co-functions
In trigonometry, co-functions are pairs of functions (like sine and cosine, tangent and cotangent, secant and cosecant) where the value of one function at an angle equals the value of its co-function at the complementary angle. Their derivatives often share a pattern, specifically, the derivatives of cotangent, cosecant, and cosine are negative.
Example:
The derivative of is negative cosecant squared, illustrating the 'negative derivative' pattern often seen with \textbf{co-functions}.
Curve Sketching
Curve sketching uses calculus concepts like derivatives to analyze the shape and behavior of a function's graph, including identifying critical points, intervals of increase/decrease, concavity, and inflection points. Derivatives of trigonometric functions are essential for sketching graphs of trig functions.
Example:
Using the first and second derivatives of helps in \textbf{curve sketching} to identify its vertical asymptotes and points of inflection.
Derivative of cot(x)
The derivative of the cotangent function, cot(x), is negative cosecant squared of x, denoted as -csc²(x). Remember that co-functions often have negative derivatives.
Example:
To find the rate of change of , you would get .
Derivative of csc(x)
The derivative of the cosecant function, csc(x), is negative cosecant x times cotangent x, denoted as -csc(x)cot(x). Like other co-functions, its derivative is negative.
Example:
Calculating the derivative of yields .
Derivative of sec(x)
The derivative of the secant function, sec(x), is secant x times tangent x, denoted as sec(x)tan(x). This derivative involves both secant and tangent functions.
Example:
If the slope of a curve is defined by , then the rate at which the slope changes is .
Derivative of tan(x)
The derivative of the tangent function, tan(x), is secant squared of x, denoted as sec²(x). This rule is fundamental for differentiating trigonometric expressions involving tangent.
Example:
If a particle's position is given by , its velocity at time would be .
Optimization
Optimization problems involve finding the maximum or minimum value of a function, often by setting its derivative to zero. Trigonometric derivatives can be used when the function to be optimized involves angles or periodic phenomena.
Example:
To find the maximum area of a sector with a fixed perimeter, you might need to use \textbf{optimization} techniques involving trigonometric functions and their derivatives.
Quotient Rule
The Quotient Rule is a formula used to find the derivative of a function that is the ratio of two other functions. If $h(x) = f(x)/g(x)$, then $h'(x) = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2$.
Example:
Finding the derivative of requires using the \textbf{Quotient Rule}.
Radians
Radians are the standard unit of angular measure used in calculus, where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius. All trigonometric derivative rules are derived and apply only when angles are measured in radians.
Example:
When evaluating at , you must first convert to \textbf{radians} to correctly apply the derivative rule.
Related Rates
Related rates problems involve finding the rate at which a quantity changes by relating it to other quantities whose rates of change are known. Derivatives of trigonometric functions frequently appear in these problems when angles or distances are involved.
Example:
A classic \textbf{related rates} problem might ask for the rate at which the angle of elevation of a rocket changes as it ascends, requiring the derivative of a trigonometric function.
Tangent Line Equation
The tangent line equation represents the line that touches a curve at a single point and has the same slope as the curve at that point. It is found using the point-slope form $y - y_1 = m(x - x_1)$, where $m$ is the derivative evaluated at $x_1$.
Example:
To find the \textbf{tangent line equation} to at , you would first calculate for the slope.
Trig Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables for which both sides of the equation are defined. They are crucial for simplifying expressions before differentiation.
Example:
Before differentiating , it's often easier to simplify it using the \textbf{trig identity} and then differentiate .