Question Video: Finding the Rate of Change of the Slope of the Tangent to a Cubic Function at a Given 𝑥-Coordinate | Nagwa Question Video: Finding the Rate of Change of the Slope of the Tangent to a Cubic Function at a Given 𝑥-Coordinate | Nagwa

Question Video: Finding the Rate of Change of the Slope of the Tangent to a Cubic Function at a Given π‘₯-Coordinate Mathematics • Third Year of Secondary School

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Find the rate of change of the slope of the tangent of function 𝑓(π‘₯) = βˆ’π‘₯Β³ at π‘₯ = 8.

02:30

Video Transcript

Find the rate of change of the slope of the tangent of function 𝑓 of π‘₯ equals negative π‘₯ cubed at π‘₯ equals eight.

We’re given that the function 𝑓 of π‘₯ is equal to negative π‘₯ cubed. The slope of the tangent to this function will be 𝑓 dash of π‘₯. 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 π‘₯. We know that 𝑓 of π‘₯ is equal to negative π‘₯ cubed. We need to differentiate to get 𝑓 dash of π‘₯ and then differentiate again to get 𝑓 double dash of π‘₯.

The general rule of differentiation states that if 𝑓 of π‘₯ is equal to π‘Ž multiplied by π‘₯ to the power of 𝑛, then 𝑓 dash of π‘₯ is equal to 𝑛 multiplied by π‘Ž multiplied by π‘₯ to the power of 𝑛 minus one. We multiply the power by the coefficient and reduce the power by one. Negative π‘₯ cubed is the same as negative one π‘₯ cubed. Three multiplied by negative one is equal to negative three. This means that 𝑓 dash of π‘₯ is equal to negative three π‘₯ squared.

In order to differentiate this again, we firstly multiply two by negative three. This is equal to negative six. 𝑓 double dash of π‘₯ is equal to negative six π‘₯ to the power of one. This is the same as negative six π‘₯. The rate of change of the slope of the tangent of the function 𝑓 of π‘₯ is equal to negative six π‘₯.

We need to work out the value of this when π‘₯ 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 π‘₯ equals negative π‘₯ cubed, then the rate of change of the slope of the tangent of this function at π‘₯ equals eight is negative 48.

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