To solve the differential equation, you first seperate the x and y variables: 

.

Then, integrate on both sides: 

 which is equal to: .

This integration was performed using the rules

 and .

(The constant 2 had no part in the integration and was therefore unchanged.) Note that there is only one constant of integration, C, in the entire equation. This is the general solution of the differential equation. We want the particular solution at the point , though. So, you plug in that point into the equation to solve for C: 

. The natural log of one is zero, so .

The final answer, which doesn't contain C, is .

The original function can be written as

Taking the first derivative using chain rule and simplifying gives us

Now, to take the second derivative, we may either use the product rule or the quotient rule. The product rule is used arbitrarily. Either of these methods will also implement the chain rule on the  term.

Simplying and finding a common denominator gives us

In the problem above, we can consider  to be a composite function , where  and .  

According to the Chain Rule, .

Applying this rule to our composite function:

. 

In the problem above, we can consider  to be a composite function , where  and . .  

According to the Chain Rule, .

Applying this rule to our composite function:

. 

In the problem above, we can consider  to be a composite function , where  and .  

According to the Chain Rule, .

Applying this rule to our composite function:

. 

We can solve this by method of separating variables:

So we integrate to get

So now we plug in the initial value  to get the specific solution:

So we solve for  to get . We plug this  into  to get

which becomes

after simplifying.

We can use separation of variables to solve the differential equation

We have 

So the general solution is , or  for . 

First, multiply both sides of the equation by dx to find

Second, integrate both sides of the equation

To integrate the right side, we can use u-substitution. Let u=sec(x). Then du=sec(x)tan(x)dx. Therefore we can rewrite the integral in terms of u as:

Next take the antiderivative of both sides

Substitute sec(x) for u to find the final solution in terms of x.

To solve this separable differential equation, we first need to rewrite it so that the left side is expressed entirely in terms of y and the right side in terms of x.

First factor the right side of the equation:

Next, divide both sides of the equation by y and multiply both sides of the equation by dx, which results in:

Integrate both sides of the equation:

Finally, exponentiate both sides of the equation to solve for y:

Finally,  is a constant, so just write it as 'c'.

Computation of the derivative requires use of the Product Rule and the Chain Rule.

A good way to remember the Product Rule is by memorizing this saying: "First Times the Derivative of the Second, Plus the Second Times the Derivative of the First." Or if that doesn't help then you can just write out the formula:

For:

 Where f(x) and g(x) are differentiable functions

As you can see, the "saying" from above matches the formula.

In this case:

 , 

Applying the Product Rule:

To compute the derivatives of  and , simply apply the chain rule:

For:

Where a is an integer and f(x) is a differentiable function.

Applying the Chain Rule:

Simplify the expression to match one of the answer choices: