Derivatives of trigonometric functions - Question

In summary: So, u'=\frac{-4sin(4x)}{cos(4x)}=-4tan(4x)Now, for v' all you need to do is to apply the power rule. So, v=x^2 and v'=2x.So, putting all this into the quotient rule you gety'=\frac{2x(-4sin(4x))-(-4tan(4x))x^2}{x^4}=\frac{-8xsin(4x)+4xtan(4x)}{x^4}Now, you just need to simplify this.In summary, to find the derivative of (ln(cos4x)) / 12x^2, you need
  • #1
Joe_K
33
0

Homework Statement





Find the Derivative of:

(ln(cos4x)) / 12x^2

Homework Equations



y' ln(x) = 1/x


The Attempt at a Solution



I have determined the correct answer, but I am still confused as to how I came to the solution. Starting with the numerator, the derivative of cos is -sin. This would produce ln(-4sin(4x)) in the numerator. Now, to take the derivative of the natural log, I would do ln(x)= 1/x, correct? So 1/-4sin(4x)?

Why does the final numerator come out to -4tan(4x)? I know I am missing a basic rule or overlooking something... but why does it become tangent?

Thank you
 
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  • #2
Joe_K said:

Homework Statement





Find the Derivative of:

(ln(cos4x)) / 12x^2

Homework Equations



y' ln(x) = 1/x


The Attempt at a Solution



I have determined the correct answer, but I am still confused as to how I came to the solution. Starting with the numerator, the derivative of cos is -sin. This would produce ln(-4sin(4x)) in the numerator. Now, to take the derivative of the natural log, I would do ln(x)= 1/x, correct? So 1/-4sin(4x)?

Why does the final numerator come out to -4tan(4x)? I know I am missing a basic rule or overlooking something... but why does it become tangent?

Thank you

You would NOT get ln(-4sin(4x)) in the numerator. You are forgetting the Chain Rule: [f(g(x))]' = f '(g(x) )* g '(x). Apply this to f(.) = ln (.) and g(x) = cos(4x).

RGV
 
  • #3
Joe_K said:
Find the Derivative of:

(ln(cos4x)) / 12x^2

First of all, this is a quotient so you need to use the quotient rule.

[tex]y=\frac{u}{v}[/tex] where u and v are functions of x, then [tex]y'=\frac{v'u-u'v}{v^2}[/tex]

Now, only tricky part there would be to determine u'. Since u=ln(cos(4x)) then we need to apply the chain rule, and Ray Vickson has already shown how this should be done.
 

FAQ: Derivatives of trigonometric functions - Question

What are derivatives of trigonometric functions?

Derivatives of trigonometric functions refer to the rate of change or slope of a trigonometric function at a specific point. It is a mathematical concept used to find the rate of change of a function with respect to its independent variable.

What is the general formula for finding derivatives of trigonometric functions?

The general formula for finding derivatives of trigonometric functions is: d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(tan x) = sec^2x, d/dx(cot x) = -csc^2x, d/dx(sec x) = sec x tan x, and d/dx(csc x) = -csc x cot x.

What are the basic rules for finding derivatives of trigonometric functions?

The basic rules for finding derivatives of trigonometric functions are the same as the general rules for finding derivatives. These include the power rule, product rule, quotient rule, and chain rule. In addition, the special trigonometric identities such as the double angle, half angle, and sum/difference formulas can also be used to simplify the process of finding derivatives of trigonometric functions.

How are derivatives of trigonometric functions used in real-life applications?

Derivatives of trigonometric functions are used in various fields such as physics, engineering, economics, and astronomy. For example, in physics, derivatives of trigonometric functions are used to calculate the velocity and acceleration of an object in motion. In engineering, they are used to design structures and machines that require precise angles and movements. In economics, they are used to analyze market trends and make predictions. In astronomy, they are used to calculate the position and movement of celestial bodies.

What are some common mistakes to avoid when finding derivatives of trigonometric functions?

Some common mistakes to avoid when finding derivatives of trigonometric functions include forgetting to use the chain rule, forgetting to simplify the expression using trigonometric identities, and confusing the derivatives of trigonometric functions with their inverse functions. It is also important to pay attention to the domain and range of the original function, as this can affect the validity of the derivative.

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