Derivative of the natural log

In summary: This is also wrong. What you meant to say was something like\frac{tan}{\sec^2 x}or\frac{tan}{(sec^2 x)}or\frac{tan}{\sec(x)}or\frac{tan}{x}or\frac{tan}{x^2}or\frac{tan}{x^3}or\frac{tan}{x^4}or\frac{tan}{x^5}or\frac{
  • #1
trajan22
134
1

Homework Statement


find dy/dx for y=ln(sec^(2)x)


Homework Equations


none


The Attempt at a Solution


1.)1/(sec(x)^2)*dy/dx sec(x)^2

2.)tan(x)/sec(x)^2

3.)i put it in terms of sin and cos...
sin(x)/cos(x)*cos(x)^2/1

4.)i canceled the cos(x) in the denominator and came out with sin(x)cos(x)
the answer i am supposed to get is 2tan(x). where did i go wrong?
 
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  • #2
The derivative of (sec(x))^2 is NOT tan(x).

Edit: Btw, 1/(sec(x)^2)*dy/dx sec(x)^2, should actually be (1/(sec(x)^2))*d(sec(x)^2)/dx
 
  • #3
Solving by chain rule, you get-:

d(ln(sec^2(x)))/d(sec^2(x)) * d(sec^2(x))/d(sec x) * d(sec x)/d(x)

Solving finally, you get 2tanx
 
  • #4
oh, right sorry about that on the edit.
and what is the derivative of (sec(x))^2, because i know that the derivative of tan(x) is (sec(x))^2 so i assumed it would work the other way around
 
  • #5
trajan22 said:
oh, right sorry about that on the edit.
and what is the derivative of (sec(x))^2, because i know that the derivative of tan(x) is (sec(x))^2 so i assumed it would work the other way around
no,the other way back is by integrating the derivative to get the original function
derivative of (sec x)^2 is 2secx.secxtanx
 
  • #6
oh ok i get it now, thanks for your help. easy mistake
 
  • #7
Notational fix:

1.)1/(sec(x)^2)*dy/dx sec(x)^2

This is wrong. What you meant to say was something like

[tex]\frac{1}{\sec^2 x} \cdot \frac{d}{dx}\left( \sec^2 x \right)[/tex]

or

[tex]\frac{1}{\sec^2 x} \cdot \frac{d(\sec^2 x)}{dx}[/tex]

or

[tex]\frac{1}{\sec^2 x} \cdot \frac{du}{dx}
\quad \quad (u = \sec^2 x)[/tex]

(In particular, you did not mean to put a y in there. You've already defined [itex]y=\ln \sec^2 x[/itex], and that's certainly not what you wanted in this particular expression)
 

1. What is the derivative of ln(x)?

The derivative of ln(x) is 1/x.

2. How do you find the derivative of ln(u), where u is a function of x?

The derivative of ln(u) is 1/u * u', where u' represents the derivative of u with respect to x.

3. Is there a special rule for finding the derivative of ln(x)?

Yes, the derivative of ln(x) follows the rule of the natural logarithm, which states that the derivative of ln(x) is 1/x.

4. Can the derivative of ln(x) be negative?

Yes, the derivative of ln(x) can be negative for values of x less than 1. This is because ln(x) is a decreasing function for x < 1, meaning its slope is negative.

5. What is the significance of the derivative of ln(x)?

The derivative of ln(x) is important in calculus and other mathematical applications. It is used to find the rate of change of a function or to solve optimization problems.

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