Derivatives - High School Math
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The speed of a car traveling on the highway is given by the following function of time:
Note that
What does this mean?
The speed of a car traveling on the highway is given by the following function of time:
Note that
What does this mean?
The function 
gives you the car's speed at time 
. Therefore, the fact that 
means that the car's speed is 
at time 
. This is equivalent to saying that the car is not moving at time 
. We have to take the derivative of 
to make claims about the acceleration.
The function
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The speed of a car traveling on the highway is given by the following function of time:
Consider a second function:
What can we conclude about this second function?
The speed of a car traveling on the highway is given by the following function of time:
Consider a second function:
What can we conclude about this second function?
Notice that the function 
is simply the derivative of 
with respect to time. To see this, simply use the power rule on each of the two terms.
Therefore, 
is the rate at which the car's speed changes, a quantity called acceleration.
Notice that the function
Therefore,
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Find 
if the function 
is given by
Find
To find the derivative at 
, we first take the derivative of 
. By the derivative rule for logarithms,
Plugging in 
, we get
To find the derivative at
Plugging in
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Find the derivative of the following function at the point 
.
Find the derivative of the following function at the point
Use the power rule on each term of the polynomial to get the derivative,
Now we plug in 
Use the power rule on each term of the polynomial to get the derivative,
Now we plug in
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Let 
. What is 
?
Let
We need to find the first derivative of f(x). This will require us to apply both the Product and Chain Rules. When we apply the Product Rule, we obtain:
In order to find the derivative of 
, we will need to employ the Chain Rule.
We can factor out a 2x to make this a little nicer to look at.
Now we must evaluate the derivative when x = 
.
The answer is 
.
We need to find the first derivative of f(x). This will require us to apply both the Product and Chain Rules. When we apply the Product Rule, we obtain:
In order to find the derivative of
We can factor out a 2x to make this a little nicer to look at.
Now we must evaluate the derivative when x =
The answer is
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Find the derivative of the following function:
Find the derivative of the following function:
We use the power rule on each term of the function.
The first term
becomes
.
The second term
becomes
.
The final term, 7, is a constant, so its derivative is simply zero.
We use the power rule on each term of the function.
The first term
becomes
The second term
becomes
The final term, 7, is a constant, so its derivative is simply zero.
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Give the average rate of change of the function 
on the interval 
.
Give the average rate of change of the function
The average rate of change of 
on interval 
is
Substitute:
The average rate of change of
Substitute:
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What is the derivative of 
?
What is the derivative of
To get 
, we can use the power rule.
Since the exponent of the 
is 
, as 
, we lower the exponent by one and then multiply the coefficient by that original exponent:
Anything to the 
power is 
.
To get
Since the exponent of the
Anything to the
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What is the derivative of 
?
What is the derivative of
To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.
Remember that anything to the zero power is one.
To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.
Remember that anything to the zero power is one.
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What is the derivative of 
?
What is the derivative of
To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.
We're going to treat 
as 
, as anything to the zero power is one.
That means this problem will look like this:
Notice that 
, as anything times zero is zero.
Remember, anything to the zero power is one.
To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.
We're going to treat
That means this problem will look like this:
Notice that
Remember, anything to the zero power is one.
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What is the derivative of 
?
What is the derivative of
To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.
We're going to treat 
as 
, as anything to the zero power is one.
Notice that 
, as anything times zero is zero.
To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.
We're going to treat
Notice that
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To solve this equation, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.
We're going to treat 
as 
since anything to the zero power is one.
 "\frac{\mathrm{d} }{\mathrm{d} x}(6x^2+8)=(2*6x^{2-1})+(0*8x^{0-1})")
Notice that 
since anything times zero is zero.
To solve this equation, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.
We're going to treat
Notice that
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What is the derivative of 
?
What is the derivative of
To solve this equation, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.
We're going to treat 
as 
since anything to the zero power is one.
Notice that 
since anything times zero is zero.
That leaves us with 
.
Simplify.
As stated earlier, anything to the zero power is one, leaving us with:
To solve this equation, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.
We're going to treat
Notice that
That leaves us with
Simplify.
As stated earlier, anything to the zero power is one, leaving us with:
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What is the derivative of 
?
What is the derivative of
To solve this equation, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.
We're going to treat 
as 
since anything to the zero power is one.
 "\frac{\mathrm{d} }{\mathrm{d} x}(12x^2+13x+4)=(2*12x^{2-1})+(1*13x^{1-1})+(0*4x^{0-1})")
Notice that 
since anything times zero is zero.
Just like it was mentioned earlier, anything to the zero power is one.
To solve this equation, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.
We're going to treat
Notice that
Just like it was mentioned earlier, anything to the zero power is one.
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What is the derivative of 
?
What is the derivative of
To take the derivative of this equation, we can use the power rule. The power rule says that we lower the exponent of each variable by one and multiply that number by the original exponent.
Simplify.
Remember that anything to the zero power is equal to one.
To take the derivative of this equation, we can use the power rule. The power rule says that we lower the exponent of each variable by one and multiply that number by the original exponent.
Simplify.
Remember that anything to the zero power is equal to one.
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What is the derivative of 
?
What is the derivative of
To take the derivative of this equation, we can use the power rule. The power rule says that we lower the exponent of each variable by one and multiply that number by the original exponent.
We are going to treat 
as 
since anything to the zero power is one.
Notice that 
since anything times zero is zero.
Simplify.
As stated before, anything to the zero power is one.
To take the derivative of this equation, we can use the power rule. The power rule says that we lower the exponent of each variable by one and multiply that number by the original exponent.
We are going to treat
Notice that
Simplify.
As stated before, anything to the zero power is one.
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What is the derivative of 
?
What is the derivative of
To find the first derivative, we can use the power rule. We lower the exponent on all the variables by one and multiply by the original variable.
Anything to the zero power is one.
To find the first derivative, we can use the power rule. We lower the exponent on all the variables by one and multiply by the original variable.
Anything to the zero power is one.
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What is the derivative of 
?
What is the derivative of
To find the first derivative, we can use the power rule. We lower the exponent on all the variables by one and multiply by the original variable.
We're going to treat 
as 
since anything to the zero power is one.
For this problem that would look like this:
Notice that 
since anything times zero is zero.
To find the first derivative, we can use the power rule. We lower the exponent on all the variables by one and multiply by the original variable.
We're going to treat
For this problem that would look like this:
Notice that
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What is the derivative of 
?
What is the derivative of
To find the first derivative, we can use the power rule. To do that, we lower the exponent on the variables by one and multiply by the original exponent.
We're going to treat 
as 
since anything to the zero power is one.
 "\frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x^2+5)=(3*x^{3-1})+(2*2x^{2-1})+(0*5x^{0-1})")
Notice that 
since anything times zero is zero.
To find the first derivative, we can use the power rule. To do that, we lower the exponent on the variables by one and multiply by the original exponent.
We're going to treat
Notice that
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What is the first derivative of 
?
What is the first derivative of
To find the first derivative for this problem, we can use the power rule. The power rule states that we lower the exponent of each of the variables by one and multiply by that original exponent.
Remember that anything to the zero power is one.
To find the first derivative for this problem, we can use the power rule. The power rule states that we lower the exponent of each of the variables by one and multiply by that original exponent.
Remember that anything to the zero power is one.
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