More questions on - finding dy/dx
1) y = sqrt 4 + sin 10 x - I am lost again
2) y = 10/sin x
answer: - 10 csc x cot x
3) y = sin x/ 8x
1) can't decide what is under the square root, use brackets
the way it stands, dy/dx = 0 + 10cosx
(looks too easy)
2) correct
3) use quotient rule
dy/dx = (8x(cosx) - 8sinx)/(64x^2)
take it from there
y = sqrt (4 + sin 10x)
Sure, I can help you find the derivative of each of these functions.
1) To find the derivative of y = sqrt(4 + sin(10x)), we can use the chain rule. The chain rule states that if we have a composition of functions, such as f(g(x)), the derivative is given by f'(g(x)) * g'(x).
Here, our outer function is the square root function, and our inner function is (4 + sin(10x)). So, we can let u = 4 + sin(10x). Taking the derivative of u with respect to x, we get du/dx = 10cos(10x).
Now, let's find dy/du. Since y = sqrt(u), the derivative dy/du is 1/(2sqrt(u)).
Finally, applying the chain rule, we have dy/dx = (dy/du) * (du/dx) = [(1/(2sqrt(u))) * 10cos(10x)]. Substituting u back in, we get dy/dx = 10cos(10x) / (2sqrt(4 + sin(10x))).
2) For y = 10/sin(x), we can simplify it using trigonometric identities.
Start by writing y as y = 10 * (1/sin(x)). Using the reciprocal identity, we know that 1/sin(x) is equal to csc(x), so we have y = 10csc(x).
Now, to find the derivative of y, we can differentiate csc(x). The derivative of csc(x) is -csc(x)cot(x).
So, the derivative of y = 10csc(x) is dy/dx = -10csc(x)cot(x).
3) For y = sin(x) / (8x), we can apply the quotient rule to find the derivative.
The quotient rule states that if we have a function of the form f(x)/g(x), the derivative is given by [f'(x) * g(x) - f(x) * g'(x)] / [g(x)]^2.
In this case, f(x) = sin(x) and g(x) = 8x. Taking the derivatives, we have f'(x) = cos(x) and g'(x) = 8.
Now, substituting these values into the quotient rule formula, we have dy/dx = [cos(x) * 8x - sin(x) * 8] / (8x)^2.
Simplifying this expression, we get dy/dx = [8xcos(x) - 8sin(x)] / (64x^2).
So, the derivative of y = sin(x) / (8x) is dy/dx = [8xcos(x) - 8sin(x)] / (64x^2).