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)
Find derivative
Use special triangles (as we’re working with sin and cos)
Sub the x value into derivative and operate on it to find the slope
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)
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
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)
Find derivative
Use special triangles (as we’re working with sin and cos)
Sub the x value into derivative and operate on it to find the slope
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)
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