# Calculus 1 : How to find differentiable of rate

## Example Questions

### Example Question #1 : How To Find Differentiable Of Rate

For the relation , compute  using implicit differentiation.

Explanation:

Computing  of the relation  can be done through implicit differentiation:

Now we isolate the :

### Example Question #2 : How To Find Differentiable Of 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.

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 .

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 .

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.

Explanation:

Taking the derivative of the function:

Evaluating at :

Manipulating the equation:

Allowing dx to be .01:

### Example Question #6 : 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  given , find

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

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 .