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

Question Video: Finding the First and Second Derivatives of a Function Using the Power Rule Mathematics • Third Year of Secondary School

Join Nagwa Classes

Attend live Mathematics sessions on Nagwa Classes to learn more about this topic from an expert teacher!

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.

Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

  • Interactive Sessions
  • Chat & Messaging
  • Realistic Exam Questions

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