How do you differentiate sin rootx?
By the chain rule, the derivative of sin(x1/2) will be the derivative of x1/2 multiplied by the derivative of the enclosing sine function. Thus,
y = sin(x1/2)
y' = (1/2)*(x-1/2)*cos(x1/2)
For further reading, you might want to consult your calculus book on the chain rule. Here is a site that (kind of) explains the chain rule, though it does have good examples: http://archives.math.utk.edu/visual.calculus/2/chain_rule.4/index.html
For step-by-step derivatives of functions, try Calc 101: http://calc101.com/webMathematica/derivatives.jsp
To differentiate y=sin(sin(x)) you need to use the chain rule. A common way to remember the chain rule is "derivative of the outside, keep the inside, derivative of the inside". First, you take the derivative of the outside. The derivative of sin is cos. Then, you keep the inside, so you keep sin(x). Then, you multiple by the derivative of the inside. Again, the derivative of sinx is cosx. In the end, you get y'=cos(sin(x))cos(x))
If the upper limit is a function of x and the lower limit is a constant, you can differentiate an integral using the Fudamental Theorem of Calculus. For example you can integrate Integral of [1,x^2] sin(t) dt as: sin(x^2) d/dx (x^2) = sin(x^2) (2x) = 2x sin(x^2) The lower limit of integration is 1 ( a constant). The upper limit of integration is a function of x, here x^2. The function being integrated is sin(t)
The sine rule(also known as the "law of sines") is: a/sin A = b/sin B = c/sin C where the uppercase letters represent angles of a triangle and the lowercase letters represent the sides opposite the angles (side "a" is opposite angle "A", and so on.) Sine Ratio(for angles of right triangles): Sine of an angle = side opposite the angle/hypotenuse written as sin=opp/hyp.
Do you mean sin(x2) or sin(x)2? In each case, you would apply the chain rule. The derivative of the sine function is the cosine, and the derivative with respect to x of axn is nax(n - 1). So if you mean: f(x) = sin(x2) Then: f'(x) = cos(x2) * 2x If you mean: f(x) = sin(x)2 Then: f'(x) = 2sin(x) * cos(x)