Question Video: Finding the First and Second Derivatives of a Function Using the Power Rule

Video Transcript

Find the first and second derivatives of the function 𝑓 of 𝑥 is equal to 0.003𝑥 cubed minus 0.004𝑥 to the fourth power.

The question gives us a function 𝑓 of 𝑥 which we can see is a polynomial. It wants us to find the first and second derivatives of our function 𝑓 of 𝑥. And since 𝑓 of 𝑥 is a function of 𝑥, we want to differentiate with respect to 𝑥. Let’s start by finding the first derivative of 𝑓 of 𝑥. That’s 𝑓 prime of 𝑥, which is equal to the derivative of 0.003𝑥 cubed minus 0.004𝑥 to the fourth power with respect to 𝑥.

Since this is a polynomial, we can differentiate each term by using the power rule for differentiation. We multiply by the exponent of 𝑥 and reduce this exponent by one. Applying the power rule of differentiation to each of our terms, we get three times 0.003𝑥 to the power of three minus one minus four times 0.004 times 𝑥 to the power of four minus one. And we can then evaluate this expression. We get 0.009𝑥 squared minus 0.016𝑥 cubed. So we found an expression for our first derivative of 𝑓 of 𝑥.

We now want to find an expression for our second derivative of 𝑓 of 𝑥. To do this, we’ll differentiate our first derivative of 𝑓 of 𝑥. This gives us 𝑓 double prime of 𝑥 is equal to the derivative of 0.009𝑥 squared minus 0.016𝑥 cubed. And again, we can see that this is the derivative of a polynomial. So we can do this term by term by using the power rule for differentiation. Applying the power rule of differentiation to each of our terms, we get two times 0.009𝑥 to the power of two minus one minus three times 0.016𝑥 to the power of three minus one.

And now, we can just evaluate this expression. We get 0.018𝑥 minus 0.048𝑥 squared. Therefore, we’ve shown if 𝑓 of 𝑥 is equal to 0.003𝑥 cubed minus 0.004𝑥 to the fourth power, we can find the first and second derivatives of this function by using the power rule for differentiation. We get 𝑓 prime of 𝑥 is equal to 0.009𝑥 squared minus 0.016𝑥 cubed and 𝑓 double prime of 𝑥 is equal to 0.018𝑥 minus 0.048𝑥 squared.