To find the derivative we need to use Product Rule and Power Rule. Power rule says that we take the exponent of the “x” value and bring it to the front. Then we subtract one from the exponent. To find the derivative we need to use product rule. Product rule states that we take the derivative of the first function and multiply it by the derivative of the second function and then add that with the derivative of the second function multiplied by the given first function. 

In equation this looks like:

And power rule looks like: 

We need to use power rule to find the derivatives of each individual function (g(x) and f(x))

Fortunately for this problem the derivatives are the same for each function so let's plug this into the Rule

To find the derivative we need to use power rule. Power rule says that we take the exponent of the “x” value and bring it to the front. Then we subtract one from the exponent. 

Then use power rule and you get

Calculate the derivative of the function g(t).

We need to recall a few rules in order to compute this derivative.

1) The derivative of a monomial can be found by multiplying the coefficent by the exponent, and decreasing the exponent by 1.

2) The derivative of cosine is negative sine.

3) The derivative of sine is cosine.

Put it all together to get our answer:

We cannot draw any conclusions about the derivative of  or its differentiablity; the expression we would want to talk about is, not . Notice that the denominator of the first fraction is incorrect.

Given y(z), find y'(z).

Now, we have three terms to our function, y(z). To find y'(z), we need to recall three rules. 

1) 

2) 

3) 

Using these three rules, we can solve our problem.

1) 

2) 

3)

Put all our terms together to get:

Given y(z), find y''(z).

Now, we have three terms to our function, y(z). To find y''(z), we need to first find y'(z), then we can differentiate again to get y"(z).

To find y'(z)... 

1) 

2) 

3) 

Using these three rules, we can complete the first step.

1) 

2) 

3)

Put all our terms together to get:

Now, we need to replicate the process to get y"(z).

1) Our first term will remain the same.

2) Our second term will follow the same rule and reduce again.

3) Now we need to recall another rule:

So now we can put it all back together to get:

The derivative of the function is equal to

and was found using the following rules:

, , , 

For a sum, the derivative is simply the sum of the derivatives of the individual parts. 

The first derivative of the function is equal to

and was found using the following rules:

, , , , , 

The derivative of the function is equal to

and was found using the following rules:

, , 

Taking the derivative without simplification gets us

and algebra is used to simplify to the final expression above.

This is a chain rule using trigonometric functions.

Which upon simplifying is: