Hi. I need help with finding the derivative of the function:
y= [(sec^7 x)/7] - [(sec^5 x)/5]
Thanks!
recall that if y = secx
dy/dx = secx tanx
y= [(sec^7 x)/7] - [(sec^5 x)/5]
dy/dx = (1/7)(7)(sec^6 x)(secx)tanx - (1/5)(5)(sec^4 x)secx tanx
= sec^7 x tanx - sec^5 x tanx
= sec^5 x tanx(sec^2 x - 1) but sec^2 x - 1 = tan^2x
= (sec^5 x)(tan^3 x)
nice question
Thank you Reiny!
I'm a little confused though.. Where did you get the (1/7) and (1/5) in the second line?
the original fractions had 7 and 5 in the denominator
which is the same as (1/7) times ....
e.g
9/4 = (1/4)(9)
Ohh I see. Thank you!
Certainly! To find the derivative of the given function y = [(sec^7 x)/7] - [(sec^5 x)/5], you can use the power rule and the chain rule of derivatives.
Here's a step-by-step explanation of how to find the derivative:
Step 1: Rewrite the expression using the power rule:
y = (1/7)*(sec x)^7 - (1/5)*(sec x)^5
Step 2: Apply the power rule to each term separately:
For the first term, (1/7)*(sec x)^7, use the power rule. The derivative of (sec x)^n with respect to x is n*(sec x)^(n-1) * sec x * tan x.
So, the derivative of (1/7)*(sec x)^7 is (1/7) * 7 * (sec x)^6 * sec x * tan x = (sec x)^6 * sec x * tan x.
For the second term, (1/5)*(sec x)^5, again use the power rule. The derivative of (sec x)^n with respect to x is n*(sec x)^(n-1) * sec x * tan x.
So, the derivative of (1/5)*(sec x)^5 is (1/5) * 5 * (sec x)^4 * sec x * tan x = (sec x)^4 * sec x * tan x.
Step 3: Simplify the derivatives obtained in step 2:
The derivative of the first term is (sec x)^6 * sec x * tan x = (sec x)^7 * tan x.
The derivative of the second term is (sec x)^4 * sec x * tan x = (sec x)^5 * tan x.
Step 4: Combine the derivatives to find the derivative of the function:
y' = (sec x)^7 * tan x - (sec x)^5 * tan x
So, the derivative of the given function y = [(sec^7 x)/7] - [(sec^5 x)/5] is y' = (sec x)^7 * tan x - (sec x)^5 * tan x.
I hope this explanation helps! Let me know if you have any further questions.