Understanding the limiting process. - AP Calculus AB

First find the derivative of the function.

f(x) = –x2 + 5√(x)

f'(x) = –2x + 5(1/2)x–1/2

Simplify the problem

f'(x) = –2x + (5/2x1/2)

Plug in 9.

f'(3) = –2(9) + (5/2(9)1/2)

= –18 + 5/(6)

Rewrite problem.

(x + 1)/(x – 1)

Use quotient rule to solve this derivative.

((x – 1)(1) – (x + 1)(1))/(x – 1)2

(x – 1) – x – 1)/(x – 1)2

–2/(x – 1)2

Need to use the power rule which states: $$\frac{d}{dx}$u^n$$=nu^{n-1}$$\frac{du}{dx}$

In our problem $\frac{du}{dx}$=-3sin(3x)

The answer is . It is easy to solve if we multiply everything together first before taking the derivative.

The correct answer is .

We must use the product rule to solve. Remember that the derivative of is .

This question is asking to evaluate a one-sided equation of a function. Specifically, the limit of the function to the left of . When is substituted into the function the result is indeterminate. This means it is in the form zero over zero.

Since the question is looking for a one-sided limit, let us substitute in a value that is slightly less than .

Use the chain rule and the formula

We will have to find the first derivative of with respect to using implicit differentiation. Then, we can find , which is the second derivative of with respect to .

We will apply the chain rule on the left side.

We now solve for the first derivative with respect to .

In order to get the second derivative, we will differentiate with respect to . This will require us to employ the Quotient Rule.

We will replace with .

But, from the original equation, . Also, if we solve for , we obtain .

The answer is .

The derivative of is equal to therefore the first part of the equation remains the same.

The second part requires regular differential rules.

Therefore when differentiating you get .

Combining the first and second part we get the final derivative:

.

The rule for taking the derivative of .

For this problem we need to remember to use the Chain Rule.

Since we are taking the derivative of,

we need to take the derivative of the outside piece keeping the inside piece the same , and then multiply the whole thing by the derivative of the inside piece .

Therefore the solution becomes:

,

.

The derivative of is .

Therefore the integral of

where C is some constant.

The derivative of,

.

The derivative of

Therefore, using the Chain Rule the derivative of the function will become the derivative of the outside piece keeping the original inside piece. Then multiplying that by the derivative of the inside piece.

Using the power rule, multiply the coefficient by the power and subtract the power by 1.

Use the product rule:

Use the product rule to find the derivative of the function.

The derivative of any function of e to any exponent is equal to the function multiplied by the derivative of the exponent.

Factoring out an x gives you .

For , the nth derivative is . As an example, consider . The first derivative is , the second derivative is , and the third derivative is . For the question being asked, the 99th derivative of would be . The 66th derivative of would be , and any higher derivative would be zero, since the derivative of any constant is zero. Thus, for the given function, the 99th derivative is .

, meaning is increasing when .

, meaning is concave up when .