Note that rules (3) to (6) can be proven using the quotient rule along with the given function expressed in terms of the sine and cosine functions, as illustrated in the following example.Įxample 1: Use the definition of the tangent function and the quotient rule to prove if f( x) = tan x, than f′( x) = sec 2 x.Įxample 3: Find if f( x) = 5 sin x + cos x.Įxample 4: Find the slope of the tangent line to the curve y = sin x at the point (π/2,1)īecause the slope of the tangent line to a curve is the derivative, you find that y′ = cos x hence, at (π/2,1), y′ = cos π/2 = 0, and the tangent line has a slope 0 at the point (π/2,1). If f( x) = csc x, then f′( x) = −csc x cot x If f( x) = sec x, then f′( x) = sec x tan xĦ. If f( x) = cot x, then f′( x) = −csc 2 x.ĥ. If f( x) = tan x, then f′( x) = sec 2 xĤ. The six trigonometric functions also have differentiation formulas that can be used in application problems of the derivative. Volumes of Solids with Known Cross Sections.Second Derivative Test for Local Extrema.First Derivative Test for Local Extrema.Differentiation of Exponential and Logarithmic Functions.
Differentiation of Inverse Trigonometric Functions.Limits Involving Trigonometric Functions.
