Calculus 1 : Differential Functions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #151 : How To Find Differential Functions

Differentiate the function:

Possible Answers:

Correct answer:

Explanation:

Take the derivative of each term in the polynomial using the power rule  where n is the exponent:

 

Example Question #341 : Differential Functions

Differentiate the function:

Possible Answers:

Correct answer:

Explanation:

Take away the parentheses by multiplying the outside piece by each term within the parentheses so that you are left with a polynomial. 

Then take the derivative of each term in the polynomial using the power rule  where n is the exponent:

Example Question #343 : Differential Functions

A company wants to find the smallest amount of material to build a box with a square bottom of volume .  Find the height of such a box.

Possible Answers:

None of the above.

Correct answer:

Explanation:

To solve this problem, we must first know that the surface area of a box with a square bottom would be  where  is the length of the side of the box and  is the height of the box.

The volume of the box can be found with the formula  where  is the length of the side of the box and  is the height of the box.

 

We know that the volume of such a box must be equivalent to 6,250; therefore to solve this problem we must first solve one of the variables in terms of another.

Then we must plug it into the surface area equation.

 

Then, take the derivative and set it equal to zero to find the minimum. To take the derivative of that problem, one must know the power rule 

.

Solving for 

 

Plugging back into the original volume equation to solve for .

 

Example Question #342 : Differential Functions

Differentiate the equation. 

Possible Answers:

None of the above.

Correct answer:

Explanation:

To approach this problem you msut first understand the different rules of calculus.  

First is the Quotient rule, 

.  

Second is the product rule, 

.  

Third is the power rule, 

.  

Fourth is the chain rule,

.  

Lastly, some identities must be known these are  and .

 

The step by step solution of the equation above is as follows

First we apply the quotient rule and power rule.

Next the product rule and chain rule as well as the identities to take the derivative of , giving our final answer.

 

Example Question #152 : How To Find Differential Functions

Given that  find 

Possible Answers:

Correct answer:

Explanation:

The first few derivatives can be evaluated to give a general trend:

Thus:

Therefore when :

Example Question #341 : Differential Functions

Differentiate:

Possible Answers:

Correct answer:

Explanation:

To differentiate this function, you must use the chain rule. To do that, you first multiply the exponent by the coefficient in front of the expression. In this case, it's one. Then, you subtract one from the exponent, so at this point, you have .

The last step is to take the derivative of what's inside the parantheses, which is .

Multiply like terms so that your derivative is

Example Question #1372 : Calculus

Differentiate:

.

Possible Answers:

Correct answer:

Explanation:

It might be helpful to rewrite this equation first: . Then, apply the chain rule.

First, multiply the exponent by the coefficient in front of the expression (in this case, 1).

Then, subtract 1 from the exponent.

Lastly, take the derivative of the expression.

This comes out to

.

Then, multiply like terms and get rid of the negative exponent:

.

Example Question #161 : Other Differential Functions

Find   given the following function  .

Possible Answers:

Correct answer:

Explanation:

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.

 

Example Question #349 : Differential Functions

Find the first derivative of the following function.

Possible Answers:

None of these

Correct answer:

Explanation:

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

Example Question #161 : How To Find Differential Functions

Derive the derivative of .

Possible Answers:

None of these

Correct answer:

Explanation:

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 .

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