### Video Transcript

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

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

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

We now want to find an expression for our second derivative of π of π₯. To do this, weβll differentiate our first derivative of π of π₯. This gives us π double prime of π₯ is equal to the derivative of 0.009π₯ squared minus 0.016π₯ 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 0.009π₯ to the power of two minus one minus three times 0.016π₯ to the power of three minus one.

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