12-05: Special Derivatives (Sinusoidal and Exponential Functions)

IROC of Sinusoidal Functions

Sinusoidal function: functions where graphs have the same shape as a sine wave

Properties of sinusoidal functions:

y=sinx and its derivative

  • Derivative of y=sinx is y=cosx

This is because the slope of the tangent line gets “derivative-d” and then becomes a point

In the original, mtangent=1 when increasing, this becomes the new y value

→ Otherwise, consider the points where mtangent=0, these become intercepts as we learned before where min and max points become intercepts when deriving

y=cosx and its derivative

Same thing as above, but this time it becomes y=-sinx

  • Derivative of y=cosx is y=-sinx

Pattern: The cycle of derivatives with sinusoidal functions

%%f(x) = sinx%%

%%f’(x) = cosx%%

%%f’’(x) = -sinx%%

%%f’’’(x) = -cosx%%

%%f’’’’(x) = sinx%%

  • Pattern repeats over and over again
  • The rate of change of a sinusoidal function is periodic
  • The derivative of a sinusoidal function is a sinusoidal function
    • Degrees aren’t changing, it will forever remain a sinusoidal function

Finding the equation of a line that is tangent to a given function and passes through a point (or has a given x value as well)

  1. Find derivative
  2. Use special triangles (as we’re working with sin and cos)
  3. Sub the x value into derivative and operate on it to find the slope
  4. Use y=mx+b to find b (sub in (x,y) that is given; if it’s not given and you just have the x value, then sub into original equation to get the y coordinate so you have a full coordinate point)
  5. Find your final equation

Derivatives of the Sine and Cosine Functions and Differentiation Rules for Sinusoidal Functions

  • f(x) = sinx → f’(x) = cosx
  • f(x) = cosx → f’(x) = -sinx

The power, chain, and product differentiation rules also apply to sinusoidal functions

Review of Exponential and Logs

Log functions are the inverse of exponential functions

Log rules:

Exponential growth and decay

Rates of Change of Exponential Functions and the Number e

As x→∞ the IROC (slope of the tangent) is increasing

  • Rate of change is increasing exponentially, therefore the exponent of an exponential function is also an exponential function

Euler’s number e

  • Irrational number
  • Similar in nature to π
    • IROC for the natural exponential function f(x)=eˣ ⟹ f’(x)=eˣ
    • The derivative of an exponential function is an exponential function

e is known as Euler’s number is defined as a limit

The natural log of x is defined as a log function with base e

  • ln has a log base e, a log with base of e

    Properties:

    Important rules:

Derivatives of Exponential Functions

  • @@The derivative of an exponential function is an exponential function@@
  • If y=bˣ then y’ = kbˣ where k is some constant

Derivative of an exponential function:

Differentiation Rules for Exponential Functions and Applications

Recall chain rule: h(x) = f’(g(x)) • g’(x)

  • When word problems ask for “rate”: find the derivative