Can you help me find the derivative of y= cos(x)/1+sin(x).. the / is a fraction bar.
I have a hard time finding trig derivatives because I don't really know what the derivative of cos,sin ,or tan would be honestly.
But, to even attempt this question, you MUST know the derivatives of the basic trig functions.
in this case, use the quotient rule
dy/dx = ( (1+sinx)(-sinx) - cosx(cosx) )/(1+sinx)^2
= (-sinx - sin^2 x - cos^2 x)/(1+sinx)^2
= (-sinx -1)(1+sinx)^2
= -(1+sinx)/(1+sinx)^2
= -1/(1+sinx)
Okay, I will get to familiarizing myself with the derivative of the trig functionss.
Thanks you so much, I really appreciate all the work and patience you tutors have :)
Of course! I can help you find the derivative of y= cos(x)/(1+sin(x)).
To find the derivative of this function, we can use the quotient rule of differentiation. The quotient rule states that for a function of the form f(x)/g(x), where f(x) and g(x) are differentiable functions, the derivative can be found using the following formula:
(dy/dx) = (g(x) * f'(x) - f(x) * g'(x)) / [g(x)]^2
Now, let's apply the quotient rule to find the derivative of y= cos(x)/(1+sin(x)).
First, we need to find the derivatives of cos(x) and 1+sin(x). The derivative of cos(x) is -sin(x), and the derivative of 1+sin(x) is simply cos(x).
Now, we can substitute these values into the quotient rule formula:
(dy/dx) = [(1+sin(x)) * (-sin(x)) - cos(x) * cos(x)] / [1+sin(x)]^2
Simplifying this expression further, we get:
(dy/dx) = [-sin(x) - sin(x)*sin(x) - cos^2(x)] / [1+sin(x)]^2
To simplify it even more, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1:
(dy/dx) = [-sin(x) - sin^2(x) - (1-sin^2(x))] / [1+sin(x)]^2
= [-2sin(x) - 1] / [1+sin(x)]^2
And there you have it! The derivative of y= cos(x)/(1+sin(x)) is (-2sin(x) - 1) / [1+sin(x)]^2.
If you're struggling with the derivatives of trigonometric functions like cos(x), sin(x), and tan(x), it's helpful to familiarize yourself with the derivatives of these basic trigonometric functions:
- The derivative of cos(x) is -sin(x).
- The derivative of sin(x) is cos(x).
- The derivative of tan(x) is sec^2(x), where sec(x) is the reciprocal of cos(x).
These derivatives can be memorized or referred to in derivative tables or calculus textbooks. Practice using these rules and identities, and with time, you will become more comfortable in differentiating trigonometric functions.