Video Transcript
Find the rate of change of the
slope of the tangent of function 𝑓 of 𝑥 equals negative 𝑥 cubed at 𝑥 equals
eight.
We’re given that the function 𝑓 of
𝑥 is equal to negative 𝑥 cubed. The slope of the tangent to this
function will be 𝑓 dash of 𝑥. We need to differentiate the
function. The rate of change also means
differentiate. Therefore, in order to work out the
rate of change of the slope of the tangent, we need to differentiate again. This will give us 𝑓 double dash of
𝑥. We know that 𝑓 of 𝑥 is equal to
negative 𝑥 cubed. We need to differentiate to get 𝑓
dash of 𝑥 and then differentiate again to get 𝑓 double dash of 𝑥.
The general rule of differentiation
states that if 𝑓 of 𝑥 is equal to 𝑎 multiplied by 𝑥 to the power of 𝑛, then 𝑓
dash of 𝑥 is equal to 𝑛 multiplied by 𝑎 multiplied by 𝑥 to the power of 𝑛 minus
one. We multiply the power by the
coefficient and reduce the power by one. Negative 𝑥 cubed is the same as
negative one 𝑥 cubed. Three multiplied by negative one is
equal to negative three. This means that 𝑓 dash of 𝑥 is
equal to negative three 𝑥 squared.
In order to differentiate this
again, we firstly multiply two by negative three. This is equal to negative six. 𝑓 double dash of 𝑥 is equal to
negative six 𝑥 to the power of one. This is the same as negative six
𝑥. The rate of change of the slope of
the tangent of the function 𝑓 of 𝑥 is equal to negative six 𝑥.
We need to work out the value of
this when 𝑥 equals eight. 𝑓 double dash of eight is equal to
negative six multiplied by eight. This is equal to negative 48. If the function 𝑓 of 𝑥 equals
negative 𝑥 cubed, then the rate of change of the slope of the tangent of this
function at 𝑥 equals eight is negative 48.
