Can someone please differentiate y = sin x / x for me?
What I did was put (sin x)(x)^-1 and used the product rule...
So for me it was (sin x)[-1(x)^-2] + [(x)^-1(cos x)]
Now as I'm looking at a similar homework problem I think my method may be wrong... If my method is wrong it looks like I don't really understand this too well. Help?
Your work is correct. There are often multiple ways to differentiate an expression.
To differentiate the function y = sin(x)/x, we can use the quotient rule. The quotient rule states that if we have a function in the form f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, the derivative is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
Let's apply this rule to differentiate y = sin(x)/x:
Step 1: Identify the functions g(x) and h(x):
In this case, g(x) = sin(x) and h(x) = x.
Step 2: Differentiate g(x) and h(x) separately:
g'(x) = cos(x) (derivative of sin(x) with respect to x)
h'(x) = 1 (derivative of x with respect to x, which is just the constant 1)
Step 3: Apply the quotient rule formula:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
= (cos(x) * x - sin(x) * 1) / x^2
= (xcos(x) - sin(x)) / x^2
So, the derivative of y = sin(x)/x is f'(x) = (xcos(x) - sin(x)) / x^2.
It seems like you used the product rule instead of the quotient rule, which is why you obtained a different answer. Remember to use the quotient rule when you have a fraction/division of functions.