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Example Questions
Example Question #1877 :Functions
For the relation, computeusing implicit differentiation.
Computingof the relationcan be done through implicit differentiation:
Now we isolate the:
Example Question #1 :Differentiable Rate
In chemistry, rate of reaction is related directly to rate constant.
, whereis the initial concentration
Give the concentration of a mixture with rate constantand initial concentration,seconds after the reaction began.
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
, whereis the initial concentration.
Plugging in our values,
Example Question #1879 :Functions
andare related by the function. Findatifandat.
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 #1880 :Functions
LetUse logarithmic differentiation to find.
The form of log differentiation after first "logging" both sides, then taking the derivative is as follows:
which implies
So:
Example Question #81 :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 inwith the afore mentioned relation.
Let. Findand, givenand find.
Taking the derivative of the function:
Evaluating at:
Manipulating the equation:
Allowing dx to be .01:
Which is our answer.
Example Question #82 :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 inwith the afore mentioned relation.
Let. Findgiven, find
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 #83 :Rate
The find the change of volume of a spherical balloon that is growing fromto
这是一个相关的问题。找啊f 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 #1 :How To Find Differentiable Of Rate
Find the rate of change ofat.
找啊f 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
atthe rate of change is
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