# IB Maths Studies Differentiation

1. IB Mathematical Studies SL – Derivatives

How can we find the derivative of the following functions?

Solution

IB Mathematical Studies SL – Derivatives

2. IB Mathematical Studies SL – Differentiation

How can we find the derivative of the following functions?

Solution

IB Mathematical Studies SL – Derivatives

So the derivative function will be

So, the derivative function will be

3. IB Mathematical Studies SL – Differentiation, Gradient of the tangent

How can we find the gradient of the tangent to at ?

Solution

IB Mathematical Studies SL – Derivatives, Gradient of the tangent

Firstly we have to simplify the function

Secondly, we have to calculate the derivative function

And finally plug-in the value in order to find the gradient of the tangent to the curve at this point

4. IB Mathematical Studies SL – Differentiation, Equation of the tangent

How can we find the equation of the tangent to at ?

Solution

IB Mathematical Studies SL – Derivatives, Equation of the tangent

Firstly we have to simplify the function

Secondly, we have to calculate the first derivative function

Then we have to plug-in the value in order to find the gradient of the tangent to the curve at this point

Since the contact point is .

Therefore the equation of the tangent to the curve at has equation:

5. IB Mathematical Studies SL – Differentiation, Second derivative

How can we find the second derivative of the following functions?

Solution

IB Mathematical Studies SL – Derivatives, Second derivative

Firstly we have to simplify the function

So, the first derivative function is:

and the second derivative is:

Similarly,

The first derivative function is

and the second derivative is:

6. IB Mathematical Studies SL – Differentiation, Monotonicity, Increasing Decreasing Function

How can we find the intervals where is decreasing or increasing?

Solution

IB Mathematical Studies SL – Derivatives,  Monotonicity, Increasing Decreasing Function

Firstly, we have to find the first derivative function

which is positive in the intervals

and is negative in the interval

Therefore the function is increasing for

and is increasing for