Video Transcript
If 𝑓 of 𝑥 equals four 𝑥 cubed plus two 𝑥 minus 𝑒 to the power of three 𝑥, find 𝑓 prime of two.
𝑓 prime of two means the first derivative of our function evaluated when 𝑥 is equal to two. First, then we’re going to need to find an expression for the first derivative 𝑓 prime of 𝑥 using differentiation. 𝑓 of 𝑥 is the sum of three differentiable terms. So in order to find its derivative, we can just differentiate each term separately and add these derivatives together.
To differentiate each of the first two terms, we use the power rule of differentiation. The derivative of four 𝑥 cubed is equal to four multiplied by three 𝑥 squared. The derivative of two 𝑥 is two multiplied by one. And then, we’ll need to consider how to find the derivative of an exponential function. Well, we recall that the derivative with respect to 𝑥 of 𝑒 to the power of some function 𝑔 of 𝑥 is equal to 𝑔 prime of 𝑥 multiplied by 𝑒 to the power of 𝑔 of 𝑥.
Here, our function 𝑔 of 𝑥 is just three 𝑥. And so, its derivative is three. We find then that the derivative of 𝑒 to the power of three 𝑥 is three 𝑒 to the power of three 𝑥. And as it was negative 𝑒 to the three 𝑥, we have negative three 𝑒 to the three 𝑥 in our derivative. Simplifying the coefficients gives our expression for 𝑓 prime of 𝑥. It’s equal to 12𝑥 squared plus two minus three 𝑒 to the power of three 𝑥.
Finally, we’ll need to evaluate this derivative when 𝑥 is equal to two. So we substitute two for 𝑥 throughout our derivative, giving 𝑓 prime of two is equal to 12 multiplied by two squared plus two minus three 𝑒 to the power of three times two. Two squared is four and 12 multiplied by four is 48. So we have 48 plus two minus three 𝑒 to the sixth power.
Simplifying and leaving our answer in terms of 𝑒 so that we have an exact answer gives 𝑓 prime of two is equal to 50 minus three 𝑒 to the sixth power.
