Calculus 1 : How to find differentiable of rate

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #1 : How To Find Differentiable Of Rate

For the relation , compute  using implicit differentiation. 

Possible Answers:

Correct answer:

Explanation:

Computing  of the relation  can be done through implicit differentiation:

Now we isolate the :

Example Question #82 : Rate

In chemistry, rate of reaction is related directly to rate constant

, where  is the initial concentration

Give the concentration of a mixture with rate constant  and initial concentration ,  seconds after the reaction began.  

Possible Answers:

Correct answer:

Explanation:

This is a simple problem of integration. To find the formula for concentration from the formula of concentration rates, you simply take the integral of both sides of the rate equation with respect to time. 

Therefore, the concentration function is given by

, where  is the initial concentration. 

Plugging in our values,

Example Question #3 : How To Find Differentiable Of Rate

 and  are related by the function .  Find  at  if  and   at .

Possible Answers:

Correct answer:

Explanation:

We will use the chain and power rules to differentiate both sides of this equation.

Power Rule: 

Chain Rule:

.

Applying the above rules to our function we find the following derivative.

at  and .

Therefore at 

 

Example Question #4 : How To Find Differentiable Of Rate

Let  Use logarithmic differentiation to find .

Possible Answers:

Correct answer:

Explanation:

The form of log differentiation after first "logging" both sides, then taking the derivative is as follows:

 

    which implies

 

 

So:

 

Example Question #5 : How To Find Differentiable Of Rate

We can interperet a derrivative as  (i.e. the slope of the secant line cutting the function as the change in x and y approaches zero) but these so-called "differentials" ( and ) can be a good tool to use for aproximations. If we suppose that , or equivalently . If we suppose a change in x (have a concrete value for ) we can find the change in  with the afore mentioned relation.

Let . Find  and, given  and find.

Possible Answers:

Correct answer:

Explanation:

Taking the derivative of the function:

Evaluating at :

Manipulating the equation:

Allowing dx to be .01:

Which is our answer.

Example Question #83 : Rate

We can interperet a derrivative as  (i.e. the slope of the secant line cutting the function as the change in x and y approaches zero) but these so-called "differentials" ( and ) can be a good tool to use for aproximations. If we suppose that , or equivalently . If we suppose a change in x (have a concrete value for ) we can find the change in  with the afore mentioned relation.

Let . Find  given , find 

Possible Answers:

Correct answer:

Explanation:

First, we take the derivative of the function:

evaluate the derivative at 

Manipulating the equation by solving for dy:

Assuming dx = 0.3

Example Question #7 : How To Find Differentiable Of Rate

The find the change of volume of a spherical balloon that is growing from  to 

Possible Answers:

Correct answer:

Explanation:

This is a related rate problem.  To find the rate of change of volume with respect to radius, we need to take the derivative of the volume of a sphere equation 

Then, we will plug in the relevant information.  The initial radius will be substituted in for , and , since that is the change from the initial to final radius of the balloon.

Example Question #8 : How To Find Differentiable Of Rate

Find the rate of change of  at .

Possible Answers:

Correct answer:

Explanation:

To find the rate of change of a polynomial at a point, we must find the first derivative of the polynomial and evaluate the derivative at that point.

For this problem,

the first derivative of this expression is

at  the rate of change is 

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