Video: Finding the First Derivative of Polynomial Functions

Find the first derivative of the function 𝑦 = π‘₯⁴ βˆ’ π‘₯Β³/4 βˆ’ 3π‘₯ + 6

02:06

Video Transcript

Find the first derivative of the function 𝑦 equals π‘₯ to the power of four minus π‘₯ cubed divided by four minus three π‘₯ plus six.

In order to find the first derivative we need to differentiate the function to work out 𝑑𝑦 by 𝑑π‘₯, or the gradient of that function. If any term 𝑦 is equal to π‘Ž multiplied by π‘₯ to the power of 𝑛, then 𝑑𝑦 by 𝑑π‘₯ is equal to 𝑛 multiplied by π‘Ž multiplied by π‘₯ to the power of 𝑛 minus one. We multiply the power by the coefficient and then decrease the power by one.

We’re going to differentiate each term of the function π‘₯ to the power of four minus π‘₯ cubed divided by four minus three π‘₯ plus six individually. Differentiating π‘₯ to the power of four gives us four π‘₯ cubed. We multiply the power four by one and decrease the power by one. The term π‘₯ cubed divided by four differentiates to three π‘₯ squared divided by four. Once again, we multiply the power three by one and then decrease the power by one.

The third term, negative three π‘₯, differentiates to negative three. Decreasing the power this time gives us π‘₯ to the power of zero. And anything to the power of zero is equal to one. Finally, differentiating any constant, in this case plus six, gives us zero. The first derivative of the function 𝑦 equals π‘₯ to the power of four minus π‘₯ cubed divided by four minus three π‘₯ plus six is four π‘₯ cubed minus three π‘₯ squared divided by four minus three.

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