This is an example of when product rule must be used to find the derivative, because there are two functions ( and  being multiplied together within .

By examining the inverse trigonometric function derivatives, it can be shown that . The derivatives of these two inverse functions are negatives of one another.

This question invokes the use of chain rule.

This question requires the use of chain rule. In this case, there are three functions to consider: first, ; second, the square root; and third, . In order to evaluate, the derivative of each of the three functions must be found and multiplied together.

This problem requires use of the chain rule twice; the first time addresses the inner function of  while the second accounts for the function . 

When the function is a constant to the power of a function of x, the first step in chain rule is to rewrite f(x).  So, the first factor of f(x) will be .  Next, we have to take the derivative of the function that is the exponent, or .  Its derivative is 10x-7, so that is the next factor of our derivative.  Last, when a constant is the base of an exponential function, we must always take the natural log of that number in our derivative.  So, our final factor will be .  Thus, the derivative of the entire function will be all these factors multiplied together: .

To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x.  Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'.  So, when we take the derivative of each term, we get .  The next step is to solve for y', so we put all terms containing y' on the left side of the equation: .  To get y' alone, divide both sides by -3 to get .  To simplify even further, we can factor a -2 out of the numerator and denominator and cancel them.  So, the final answer is .

To take the derivative, you must first take the derivative of the outside function, which is sine.  However, the , or the angle of the function, remains the same until we take its derivative later.  The derivative of sinx is cosx, so you the first part of  will be .  Next, take the derivative of the inside function, .  Its derivative is , so by the chain rule, we multiply the derivatives of the inside and outside functions together to get .

To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x.  Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'.  So, when we take the derivative of each term, we get   The next step is to solve for y', so we put all terms containing y' on the left side of the equation: .  To get y' alone, divide both sides by  to get .  To simplify even further, we can factor a 2 out of the numerator and denominator and cancel them.  So, the final answer is .