calculate differentiation of cos x to x using first principle of derivative. Find the value of differentiated function , if x = 270°
Nice video here of your problem
https://www.youtube.com/watch?v=V3FoNfYstNk
cos x to x ?
not cos x
what about this x = 270°?
cosxsinxcosx
cosxsinxcosx
To calculate the differentiation of cos(x) with respect to x using the first principle of derivative, we need to find the limit as h approaches 0 of the expression [cos(x + h) - cos(x)] / h.
Let's start by expanding cos(x + h) using the cosine angle sum formula: cos(x + h) = cos(x)cos(h) - sin(x)sin(h).
Substituting this into our expression, we get:
[cos(x)cos(h) - sin(x)sin(h) - cos(x)] / h
Simplifying this expression further, we have:
[(cos(x)cos(h) - cos(x)) - sin(x)sin(h)] / h
Factoring out cos(x), we get:
cos(x)(cos(h) - 1) - sin(x)sin(h) / h
Next, we can use the trigonometric limit identity: lim h→0 (sin(h) / h) = 1. This allows us to simplify the expression further:
cos(x)(cos(h) - 1) - sin(x)sin(h)
Now, we can substitute h = 0 into the expression and calculate the value when x = 270° (or π/2 radians).
cos(270°)(cos(0) - 1) - sin(270°)sin(0)
= (-1)(1 - 1) - (-1)(0)
= -1
Therefore, the value of the differentiated function at x = 270° is -1.