find f'(x)
f(x)= sin(3x)/2x^2-4
To find the derivative (f'(x)) of the given function f(x), you will need to use the quotient rule. The quotient rule states that for a function of the form f(x) = g(x) / h(x), where g(x) and h(x) are differentiable functions, the derivative f'(x) is calculated as follows:
f'(x) = [h(x) * g'(x) - g(x) * h'(x)] / [h(x)]^2
Now, let's find the derivative of the given function f(x) = sin(3x) / (2x^2 - 4).
Step 1: Identify g(x) and h(x):
g(x) = sin(3x)
h(x) = 2x^2 - 4
Step 2: Find the derivatives g'(x) and h'(x):
g'(x) = d/dx (sin(3x))
= 3cos(3x) (using the chain rule)
h'(x) = d/dx (2x^2 - 4)
= 4x (using the power rule)
Step 3: Apply the quotient rule formula:
f'(x) = [h(x) * g'(x) - g(x) * h'(x)] / [h(x)]^2
= [(2x^2 - 4) * (3cos(3x)) - (sin(3x)) * (4x)] / [(2x^2 - 4)]^2
Simplifying further is possible, but this is the final expression for the derivative f'(x) of the given function f(x).