Since F(x) equals the integral of this function,  F'(x) must then be the derivative.

The derivative is the integrand, of which we are taking the integral.

Evaluate the integrand at the bounds that you see for the integral.  Doing this will give us the value of the derivative.

First recall the power rule which states that the derivative of  is .

That makes the derivative of  equal  .

To find the derivative of the second part of the function we must use the multiplication rule. It states that the derivative of

 is .

Using trigonometric derivatives we know the derivative of sine is cosine. Also, the rule associated for the derivative of  is .

Thus the derivative of 

 is 

so the final answer is

The quotient rule states that the derivative of  is .

The rules of trigonometric functions states that the derivative of sine is cosine and the derivative of cosine is negative sine.

The derivative of tangent is thus,

By the Pythagorean trig identities the above is equal to .

The chain rule states that the derivative of a function in the form  is .

The power rule states that the derivative of  is .

As the outer function is the sine function and the inner one is the natural log we must use those derivative rules.

The derivative of sin is cos and the derivative of  is .

Thus the answer is

.

Notice that this differential equation is a "Seperable Differential Equation".  That is, we can multiply the denominator over to the right side and then integrate both sides to find the general solution. 

Seperate the variables and bring dx to the right hand side of the equation.

Remember to take any constant coefficient numbers outside the integral.

Integrate,

Recall the power rule of integration,

Therefore we get:

This problem requires multiple uses of the chain rule for derivatives:

For the equation

Take the derivative of the value in the exponent, the  term, which in turn requires taking the derivative of that function's exponent, the  term.

This gives the value , yielding the answer:

or

This derivative requires the use of the chain rule, so work from the inside outwards.

In the function:

The derivative of  is  and the derivative of  is , so the derivative of  is . That leaves the outside.

The derivative of  is  and allows the determination of the complete derivative:

To do this formula, utilize the product rule for derivatives:

For our function

Let  and 

We can find the first derivative:

Now for the second term, use the product rule for derivatives again:

Putting everything together, the derivative is:

We can use the Product Rule, which says that for a function

 (where  and  can be any function),  

.

Applying this rule to our particular problem we get the following.

The function is a product of two functions,  and , the latter of which is composed of  and .

So we use the Product Rule and the Chain Rule, respectively.

The Product Rule: for a function , .

The Chain Rule: for a function , .

Applying these rules to our particular function we get the following derivative.