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5.2 The Derivative of the General Exponential Function y=b^x

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5.2 The Derivative of the General Exponential Function y=b^x

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Key Points and Rules
-  The ratio f'(x)/f(x) is the stretch/compression factor of f'(x)
- If f(x)=bx, then f'(x)=(ln b)(bx)
- If f(x)=bg(x), then f'(x)=(bg(x))(ln b)(g'(x))

Real life applications

-Knowing how to find derivatives of exponential functions allows us to compare rates of change of functions describing radioactive decay, population growth or loan interest rates.

Procedure
Examples:

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5.3 Optimization Problems Involving Exponential Functions

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5.3 Optimization Problems Involving Exponential Functions

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Helpful video

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Optimization with Exponential Functions Grade 12 Calculus and Vectors Lesson 5 3 7 19 13)

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Uploaded by Belcastro Math on 2013-07-19.

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Chapter 5.4:  The Derivatives of y=sinx and y=cosx 

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Chapter 5.4:  The Derivatives of y=sinx and y=cosx 

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In this Chapter, we use the derivatives of y=sinx and y=cosx to solve various types of problems involving finding tangents, equations, differentiating, and creating graphs. Using the derivatives, we are able to find the slope of the graph to investigate minima and maxima.

d/dx (sinx)= cosx     read as "the derivative of y=sinx is y=cosx"
and its RULE is: If y=sin f(x), then y'= cosf(x) x f'(x)

d/dx (cosx)= -sinx     read as "the derivative of y=cosx is y=-sinx"
and its RULE is: If y=cos f(x), then y'= -sinf(x) x f'(x)

*remember to use derivative rules when differentiating!

Real Life Applications:
The graphs are used to measure:
-music (instrument vibrations and acoustics)
-ocean tides and currents

Procedure for Sample Problems:
Example: Find the Derivative
y= 2cos4x
Steps: Take the derivative of y=cosx, which is y'=-sinx. Take the derivative of 4x, which is 4. Multiply it by the coefficient in the front.
y'=-2sin4x (4)
y'=-8sin4x

Example: Differentiate the function
y=3sinx cosx
Steps: Here is where you will use the product rule. Find the derivatives of both parts of the function. Factor, and then apply trigonometric identities if possible.
y'=(3sinx)(-sinx) + (cosx)(3cosx)
y'=-3sin²x + 3cos²x
y'=3(cos²x - sin²x)
y'=3cos(2x)

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In this Chapter, we use the derivative of y=tanx to solve various types of problems such as the ones in chapter 5.4. We can use this information to find tangents, equations, graphs, and to be able to differentiate. We are also able to observe minima and maxima changes in the graph.

d/dx (tanx)= sec²x     read as "the derivative of y=tanx is y=sec²x
and its RULE is: if y=tan f(x), then y'= sec2 f(x) x f'(x)

Real Life Applications:
The graph is used to measure:
-electrical currents
-GPS/Satellite Orbital Mechanics

Procedure for Sample Problems:
Example: Find the Derivative
y=tan6x
Steps: Take the derivative of y=tanx, which is y'=sec ² x. Then take the derivative of 6x which is 6.
y'=sec²6x (6)
y'=6sec²6x

Example: Find the Derivative:
y=tan(cosx)
Steps: Take the derivatives of y=tanx and y=cosx, which are y'=sec ² x and y'=-sinx. Multiply the equation by the derivative of what is in the bracket, y'=-sinx.
y'=sec²x(cosx) (-sinx)
y'=-sinx sec²x (cosx)