- Calculate the derivatives of sine and cosine functions, including second derivatives and beyond
- Determine the derivatives for basic trig functions like tangent, cotangent, secant, and cosecant
Derivatives of the Sine and Cosine Functions
The Main Idea
- Derivative of Sine: [latex]\frac{d}{dx}(\sin x) = \cos x[/latex]
- Derivative of Cosine: [latex]\frac{d}{dx}(\cos x) = -\sin x[/latex]
- Graphical Interpretation:
- Where sine has a maximum or minimum, cosine (its derivative) is zero
- Where cosine has a maximum or minimum, negative sine (its derivative) is zero
Find the derivative of [latex]f(x)= \sin x \cos x.[/latex]
Find the derivative of [latex]f(x)=\dfrac{x}{\cos x}[/latex].
Find the derivative of [latex]f(x) = 5x^3 \sin x[/latex].
A particle moves along a coordinate axis with position [latex]s(t) = 2\sin t - t[/latex] for [latex]0 \leq t \leq 2\pi[/latex]. At what times is the particle at rest?
A particle moves along a coordinate axis. Its position at time [latex]t[/latex] is given by [latex]s(t)=\sqrt{3}t+2 \cos t[/latex] for [latex]0\le t\le 2\pi[/latex]. At what times is the particle at rest?
Derivatives of Other Trigonometric Functions
The Main Idea
- Derivatives of Other Trigonometric Functions:
- [latex]\frac{d}{dx}(\tan x) = \sec^2 x[/latex]
- [latex]\frac{d}{dx}(\cot x) = -\csc^2 x[/latex]
- [latex]\frac{d}{dx}(\sec x) = \sec x \tan x[/latex]
- [latex]\frac{d}{dx}(\csc x) = -\csc x \cot x[/latex]
- Derivation Methods:
- Use quotient rule for [latex]\tan[/latex] and [latex]\cot[/latex]
- Apply chain rule and trigonometric identities for [latex]\sec[/latex] and [latex]\csc[/latex]
Find the derivative of [latex]f(x)=2 \tan x-3 \cot x[/latex].
Find the derivative of [latex]f(x) = \csc x + x \tan x[/latex].
Find the slope of the line tangent to the graph of [latex]f(x)= \tan x[/latex] at [latex]x=\frac{\pi}{6}[/latex].
Find the equation of a line tangent to the graph of [latex]f(x) = \cot x[/latex] at [latex]x = \frac{\pi}{4}[/latex].
Higher-Order Derivatives of Trig Functions
The Main Idea
- Cyclic Pattern of Derivatives:
- For [latex]\sin x[/latex]: [latex]\sin x \to \cos x \to -\sin x \to -\cos x \to \sin x[/latex]
- For [latex]\cos x[/latex]: [latex]\cos x \to -\sin x \to -\cos x \to \sin x \to \cos x[/latex]
- Predicting Higher-Order Derivatives:
- Use the remainder when the derivative order is divided by [latex]4[/latex]
- Pattern repeats every four derivatives
Problem-Solving Strategy
- Find the order of the derivative ([latex]n[/latex])
- Calculate [latex]n \bmod 4[/latex] (remainder when [latex]n[/latex] is divided by [latex]4[/latex])
- Use the remainder to determine the derivative:
- For [latex]\sin x[/latex]:
- Remainder [latex]0[/latex]: [latex]\sin x[/latex]
- Remainder [latex]1[/latex]: [latex]\cos x[/latex]
- Remainder [latex]2[/latex]: [latex]-\sin x[/latex]
- Remainder [latex]3[/latex]: [latex]-\cos x[/latex]
- For [latex]\cos x[/latex]:
- Remainder [latex]0[/latex]: [latex]\cos x[/latex]
- Remainder [latex]1[/latex]: [latex]-\sin x[/latex]
- Remainder [latex]2[/latex]: [latex]-\cos x[/latex]
- Remainder [latex]3[/latex]: [latex]\sin x[/latex]
- For [latex]\sin x[/latex]:
For [latex]y= \sin x[/latex], find [latex]\dfrac{d^{59}}{dx^{59}}(\sin x)[/latex].
Find [latex]\dfrac{d^{74}}{dx^{74}}(\sin x)[/latex].
A block attached to a spring is moving vertically. Its position at time [latex]t[/latex] is given by [latex]s(t)=2 \sin t[/latex].
Find [latex]v\left(\frac{5\pi}{6}\right)[/latex] and [latex]a\left(\frac{5\pi}{6}\right)[/latex]. Compare these values and decide whether the block is speeding up or slowing down.