is increasing when .

To solve, simply differentiate  and plug in . Thus,

To solve the separable differential equation, we must separate the x and dx, y and dy terms, to opposite sides of the equation:

Now, integrate both sides to solve for y:

The following rules were used for integration:

, , 

Note that we combined the C's to make one constant of integration.

Finally, isolate y by itself:

In order to find the derivative of a function using the Power Rule, mulitply the constant in front of the variable (x) by the current power that the variable is raised to, then decrease that power by 1. For example, the derivative of Axⁿ is equal to (n*A)x^n-1. If there is no variable, and just a constant, then that constant will not be changing over time, and is the derivative (rate of change) will be equal to zero. 

In order to find the local maximum, the derivate of the function f(x) must be taken. This is because the derivative of f(x) gives the rate of change for f(x). When f(x) is approaching its maximum, it is increasing. When it reaches the maximum, it stays constant (at the very top before it being to decrease, there is a point where the slope of the tangent line is equal to zero). Directly after reaching the maximum, f(x) begins to decrease. Therefore, the local maximum can be found using the derivative and noting where f'(x) crosses the x-axis, known as the x-intercept. This point could be either a local maximum of minimum, depending on the slope of f'(x). If f'(x) is decreasing, it is approaching a local maximum, and if it is increasing, a local minimum.

Setting this equal to zero and solving using the quadratic formula:

This gives us x = -1.6 or x = 3.1. In order to determine if these indicate a local maximum of minimum, the second derivative will show if the slope is positive or negative at these points. If f''(x) is negative, it is a local maximum, if positive, it is a local minimum.

Therefore, the local maximum is located at x = -1.6.

The function f(x) is given by . In order to take the derivative of such a complicated function, it must be taken in steps, by working from the "outside in", and taking the derivative of each piece along the way. Below illustrates each individual step needed to find the derivative of the sin function:

1)  This is the original sine function.

2)  The 2 exponent is brought to the front using the Power Rule.

3)  The derivative of sine is then taken and multiplied by the previous step.

3)  The derivative of the sine argument is taken and multiplied by the previous step.

Therefore, the derivative of   is equal to .

To find the derivative of this function, recall the product rule. You multiply the first expression by the derivative of the second and the add that to the product of derivative of the first expression by the second. Therefore, . Then, multiply and combine like terms so that your final answer is: .

 is the correct answer. 

When taking the derivative of a function, multiply the exponent by the constant in front of  and subtract one from the exponent. 

So,  becomes  and so on maintaining the positive and negative symbols.

Here, you are required to use the chainrule because there is a function inside of a function.

Therefore, you first differentiate  as a whole treating the inside of the parenthesis as just .

You get  .

Then, you apply the chain rule and differentiate the function inside the parenthesis:

You get  .

So your full answer is: 

To find the slope of a curve at a certain point, you must first find the derivative of that curve. 

The derivative of this curve is .

Then, plug in  for  into the derivative and you will get  as the slope of  at .