IB Maths HL Derivatives
Questions and Answers
1. IB Mathematics HL – Derivatives, Differentiation from first principles
How can we find the first derivative function of from first principles?
Solution
IB Mathematics HL – Derivatives, Differentiation from first principles
We know that the definition of the first derivative function is given by the following formula:
So in our case, we have the following
Thus the first derivative function is
2. IB Math HL, Differentiation from first principles
IB Mathematics HL – Derivatives, Differentiation from first principles
How can we find the first derivative function of from first principles?
Solution
IB Mathematics HL – Derivatives, Differentiation from first principles
We know that the definition of the first derivative function is given by the following formula:
So, we have the following
Thus the first derivative function is
3. IB Math HL, Differentiation Product Rule
IB Mathematics HL – Derivatives, Differentiation Product Rule
How can we find the first derivative function of .
Solution
IB Mathematics HL – Derivatives, Differentiation Product Rule (or Leibnitz rule)
We know that the derivative of the product of two functions is given by the following formula
So about your question we notice that can be trated as a product of two functions
.
Using the product rule we obtain
4. IB Maths HL, Differentiation Quotient Rule
IB Mathematics HL – Derivatives, Differentiation Quotient Rule
How can we find the first derivative function of .
Solution
IB Mathematics HL – Derivatives, Differentiation Quotient Rule
We know that the derivative of the quotient of two functions is given by the following formula
So about your question we notice that can be treated as a quotient of two functions
.
Using the quotient rule we obtain
5. IB Maths HL, Differentiation Chain Rule
IB Mathematics HL – Derivatives, Differentiation Chain Rule
How can we find the first derivative function of ?
Solution
IB Mathematics HL – Derivatives, Differentiation Chain Rule
We know that the derivative of composite functions is given by the following formula
The chain rule states that the derivative of a function of a function (composite function) equals derivative of the outer function times derivative of the inner function.
We can view as a composite function
, where
.
Using the Chain rule we obtain
6. IB Maths HL, Chain Rule
IB Mathematics HL – Derivatives, Differentiation Chain Rule
How can we find the first derivative function of .
Solution
IB Mathematics HL – Derivatives, Differentiation Chain Rule
We know that the derivative of composite functions is given by the following formula
The chain rule states that the derivative of a function of a function (composite function) equals derivative of the outer function times derivative of the inner function.
So about your question we view as a composite function
, where
.
Using the Chain rule we obtain
7. IB Mathematics HL, Chain Rule
IB Maths HL – Derivatives, Differentiation Chain Rule
How can we find the first derivative function of
Solution
IB Mathematics HL – Derivatives, Differentiation Chain Rule
We know that the derivative of composite functions is given by the following formula
The chain rule states that the derivative of a function of a function (composite function) equals derivative of the outer function times derivative of the inner function.
So about your question we view as a composite function
, where
.
Using the Chain rule we obtain
8. IB Maths HL, Optimization problems
IB Mathematics HL – Derivatives, Differentiation, Optimization problems
How can we find the largest area of a rectangular region having perimeter 400 m ?
Solution
IB Mathematics HL – Derivatives, Differentiation, Optimization problems
Let be the dimensions of the rectangular region and
be its area. We want to find the largest value of A.
We know that . We need to eliminate
using the fact that
Thus the function described the area can be written as
Therefore the problem now is to maximize the function
To find the maximum of , we find the stationary points
Then we use the second derivative test to determine the nature of the stationary point
So, by the second derivative test, the function has a maximum when
and the largest area of a rectangular region is
and the rectangular will be a square
9. IB Math HL, Optimization problems
IB Mathematics HL – Derivatives, Differentiation, Optimization problems
How can we determine the dimensions of a cylindrical can that will hold 1.5 liliters and will minimize the amount of material used in its construction?
Solution
IB Mathematics HL – Derivatives, Differentiation, Optimization problems
Let be the radius and the height of the cylinder respectively.
Let be its area. We want to find the minimum value of A.
We know that . We need to eliminate the variable
using the fact that
Thus the function described the area can be written as
Therefore the problem now is to minimize the function
To find the minimum of , we find the stationary points
Then we use the first derivative test to determine the nature of the stationary point
It is easy to see that for all
and for all
Thus the minimum value of the area occurs when and the minimum area amount of material used in its construction is
10. IB Maths HL, Equation of Tangent
IB Mathematics HL – Derivatives, Differentiation, Tangent and Normals
How can we find the equation of the tangent to the curve at the point
.
Solution
IB Mathematics HL – Derivatives, Differentiation, Tangent and Normals
We know that the gradient of a curve at any point
is equal to the gradient of the tangent (
) to the curve at that point.
So, we have that
and the gradient of the tangent () at the point
is
and the equation of the tangent is given by the following formula
where in this case and finally, the equation of the tangent is
11. IB Maths HL, Equation of Normal
IB Mathematics HL – Derivatives, Differentiation, Tangent and Normals
How can we find the equation of the normal to the curve at the point
.
Solution
IB Mathematics HL – Derivatives, Differentiation, Tangent and Normals
We know that the gradient of a curve at any point
is equal to the gradient of the tangent to the curve at that point. The slope of the normal (
) is the negative reciprocal of the tangent’s slope (
) since the normal is perpendicular to the tangent.
So, we have that and the gradient of the tangent (
) at the point
is
thus the slope of the normal is
and the equation of the normal is given by the following formula
where in your case and finally, the equation of the tangent is
12. IB Maths HL, Tangent line
IB Mathematics HL – Derivatives, Differentiation, Tangent and Normals
How can we find the equation of the tangent to the curve that is parallel to the line with equation
.
Solution
IB Mathematics HL – Derivatives, Differentiation, Equations of Tangent and Normals
We know that the gradient of a curve at any point
is equal to the gradient of the tangent (
) to the curve at that point.
So, we have that
and the gradient of the tangent must be equal to
since is parallel to the line
and the equation of the tangent is given by the following formula
where in this case
and finally the equation of the tangent is
12. IB Maths HL, Implicit Differentiation
IB Mathematics HL – Derivatives, Implicit Differentiation
How can we find for
.
Solution
IB Mathematics HL – Derivatives, Implicit Differentiation
Sometimes functions are given not in the form but in a more complicated form in which is difficult to express y explicitly in terms of x. Such functions are called implicit functions. In these cases, we are differentiating with respect to x and for the y-terms the Chain Rule is applied.
Differentiating with respect to x:
13. IB Maths HL Implicit Differentiation
IB Mathematics HL – Derivatives, Implicit Differentiation
How can we find for
.
Solution
IB Mathematics HL –Implicit Differentiation
Sometimes functions are given not in the form but in a more complicated form in which is difficult to express y explicitly in terms of x. Such functions are called implicit functions. In these cases, we are differentiating with respect to x and for the y-terms the Chain Rule is applied.
Differentiating with respect to x:
14. IB Maths HL Stationary points
IB Mathematics HL – Derivatives, monotonicity, increasing functions, turning points
How can we find monotonicity (increasing or decreasing) and the turning points for the function .
Solution
IB Mathematics HL – Derivatives, monotonicity, increasing functions, turning points
The first derivative function is
By setting the derivative equal to zero,
So, we know that there is a stationary point when .
From the derivative we know that since when
the function is increasing for
Similarly, since when
the function is decreasing for
.
Therefore, we can deduce that the stationary point at is a local maximum.
15. IB Maths HL Second Derivative test
IB Mathematics HL – Derivatives, Turning points, 2nd derivative test
How can we find coordinates and nature of all the stationary points of the function .
Solution
IB Mathematics HL – Derivatives, monotonicity, increasing functions, turning points
We know that
If and
then at
the function has a minimum turning point
If and
then at
the function has a maximum turning point
If and
and the second derivative function changes sign around
then at
the function has a stationary point of inflection.
The first derivative function is
By setting the derivative equal to zero,
So, the x-coordinates of the stationary points are .
By substituting into the original equation
and
Thus the coordinates of the stationary points are
The second derivative function is
By substituting the values of the stationary points into the second derivative function
, so is a local minimum point and
, so is a local maximum point
16. IB Maths HL Point of inflection
IB Mathematics HL – Derivatives, 2nd derivative test and Points of inflection
How can we find if the function has a point of inflexion?
Solution
IB Mathematics HL – Derivatives, Points of inflexion
We know that
If and the second derivative function changes sign around
then at
the function has a point of inflection.
The first derivative function is
The second derivative function is
By setting the second derivative equal to zero,
To determine the set of values for which the function is concave up or concave down we need to solve the following inequality
We observe that the function is concave up when and is concave down when
Thus at the function has a point of inflexion which has coordinates (0,16).
17. Differentiation, Derivatives, gradient, tangent- IB Mathematics HL
How can we find the equation of the tangent to the curve with equation
at the point
Solution
The gradient of the curve at is
The equation of the tangent on this point is given by the equation
, where is the gradient of the tangent and the touch point is
.
So, the equation of the tangent at the given point is
18. Differentiation, Derivatives, gradient, tangent – IB Mathematics HL
How can we find the equation of the tangent to the curve with equation
at the point
?
Solution
The gradient of the curve at is
The equation of the tangent on this point is given by the equation
where is the gradient of the tangent and the touch point is
.
So, the equation of the tangent at the given point is