This week's material focused on the eighth derivative rule regarding derivatives of exponential and logarithmic functions with introduction to the number e. The material also introduced derivatives of inverse trigonometric functions.
Derivatives: "e" as an Exponential Function
In the exponential section of derivatives, we were introduced to the numerical value e. The derivative of e in an exponential function will always equal itself, as written below.
- dy/dx (ex) = ex
Applying "e" to Several Derivative Rules
*Product Rule
*Quotient Rule
*Chain Rule
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Derivatives: Exponential Functions with Values Other than "e"
The derivative of an exponential function that is not ex (i.e. ax) is:
- dy/dx (ax) = ln (a) * ax
"a" is only representative of a value. Numerical values can be substituted for this variable, and the same steps can be followed to find the derivative. An example of this process with numerical values is listed below.
*Derivative of an Exponential Function Other Than "e"
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Derivatives: Natural Log Functions
The derivative for natural log functions is given and proven as listed below:
- dy/dx [ln(x)] = 1/x
*Proving the Derivative Formula
Now we can apply this formula to natural logs of numerical values opposed to only working with variables.
*Derivative of a Natural Log Function
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Derivatives: Logarithmic Functions
Logarithmic functions have a derivative formula that follows the format listed below:
- dy/dx [loga(x)] = 1/ln (a) * x
*Derivative of a Logarithmic Function
Derivatives of Inverse Trig Functions
Aside from working with exponential and logarithmic functions, this week's material encompassed an introduction to derivatives of inverse trig functions for sine, cosine, and tangent. Each of these trig functions has a specific derivative formula.
Derivatives: Inverse Sine
The derivative formula for finding the derivative of inverse sine is provided below with an example highlighting how the formula can be applied.
- dy/dx [sin-1(x)] = 1/√(1-x2)
*Derivative of Inverse Sine Function
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Derivatives: Inverse Cosine
The formula for the derivative of inverse cosine varies slightly in comparison to the formula for the derivative of inverse sine. One important thing to note is that inverse cosine is negative and inverse sine is positive. Remember this to avoid using the wrong formula.
- dy/dx [cos-1(x)] = -1/√(1-x2)
*Derivative of Inverse Cosine Function
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Derivatives: Inverse Tangent
Finally, the derivative formula for the inverse of tangent follows:
- d/dx [tan-1(x)] = 1/1+x2
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Don't forget several log properties that are important to use:
- ln(ab) = ln(a) + ln(b)
- ln(a/b) = ln(a) - ln (b)
- ln(ax) = x * ln(a)
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