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

Find the first and second derivatives of the function 𝑓(π‘₯) = 0.003π‘₯Β³ βˆ’ 0.004π‘₯⁴.

02:22

Video Transcript

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

The question gives us a function 𝑓 of π‘₯ which we can see is a polynomial. It wants us to find the first and second derivatives of our function 𝑓 of π‘₯. And since 𝑓 of π‘₯ is a function of π‘₯, we want to differentiate with respect to π‘₯. Let’s start by finding the first derivative of 𝑓 of π‘₯. That’s 𝑓 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 𝑓 of π‘₯.

We now want to find an expression for our second derivative of 𝑓 of π‘₯. To do this, we’ll differentiate our first derivative of 𝑓 of π‘₯. This gives us 𝑓 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, we’ve shown if 𝑓 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 𝑓 prime of π‘₯ is equal to 0.009π‘₯ squared minus 0.016π‘₯ cubed and 𝑓 double prime of π‘₯ is equal to 0.018π‘₯ minus 0.048π‘₯ squared.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.