Are you certain that you have to use the limit definition of the derivative here? Have you learned the chain rule yet?Absolutism said:Homework Statement
f(x)=sin(2x+5)
Homework Equations
I am supposed to find the derivative, and the domain of it. The problem is that I cannot deal with the simplification of it.
The Attempt at a Solution
As far as I know, I am suppsosed to use f(x+h)-f(x)/h
= sin(2(x+h)+5)-(sin(2x+5))/h
I tried expanding what's inside, as in sin(2x+2h+5)-sin(2x+5)/h
But that doesn't seem to lead to a solution
Absolutism said:I have learned the chain rule, and I am certain I am supposed to use the limit definition in this question
Mark44 said:OK, then I think you'll need to use the sum identity for sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
You will probably need to use a couple of limit formulas as well:
[tex]\lim_{h \to 0}\frac{sin(h)}{h} = 1[/tex]
[tex]\lim_{h \to 0}\frac{cos(h) - 1}{h} = 0[/tex]
lurflurf said:recall the trigonometric identity
sin(2(x+h)+5)-(sin(2x+5))=2 sin(h) cos(h+2 x+5)
or in general
sin(A)-sin(B)=2 sin(A/2-B/2) cos(A/2+B/2)
and use the facts that
cosine is continuous
sin'(0)=1
Absolutism said:Am I supposed to solve for when a --> 0?
That's what I did.
sin(2x+5)-sin(2a+5) = 2sin ((x-a)/2) cos ((x+a)/2)
...
SammyS said:Actually sin(2x+5)-sin(2a+5) = 0
You need to look at sin((2x+5)+h)-sin(2a+5) = 2sin (h/2) cos ((2x+5)+h/2) .
A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It describes how the output of a function changes with respect to its input.
To find the derivative of a trigonometric function, you can use the power rule or the chain rule, depending on the specific trigonometric function. For example, the derivative of sine is cosine, and the derivative of tangent is secant squared.
The domain of a derivative is the set of all input values for which the derivative function is defined. In other words, it is the set of all values for which the rate of change of the original function can be calculated.
Finding the derivative of a function allows us to analyze the behavior of the function at a specific point and understand how it changes over a given interval. It also helps us find the maximum and minimum points of a function and determine its concavity.
Yes, the derivative of a trigonometric function can be undefined at certain points, especially when the function has sharp turns or vertical tangents. These points are known as points of discontinuity, and the derivative cannot be calculated at these points.