Video Transcript
Find the value of d squared 𝑦 by d𝑥 squared at d to the fourth power of 𝑦 by d𝑥 to the fourth is equal to zero given 𝑦 is equal to two 𝑥 raised to the power five plus 20𝑥 raised to the power four minus 57𝑥 squared minus 126𝑥.
We’re asked to find the value of the second derivative of the given function 𝑦 at the point where the fourth derivative with respect to 𝑥 is equal to zero. This means that we’ll need to differentiate our expression for 𝑦 four times, put our result equal to zero, and solve this for 𝑥 and then substitute our solution for 𝑥 back into the second derivative. Our expression for 𝑦, which is two 𝑥 raised to the power five plus 20𝑥 raised to the fourth power minus 57𝑥 squared minus 126𝑥, is a polynomial in 𝑥. In fact, it’s a quintic polynomial with the highest power of 𝑥 equal to five. And to differentiate this, we can use the power rule for derivatives.
This says for an expression 𝑦 which is 𝑎𝑥 raised to the 𝑛th power where 𝑎 and 𝑛 are constants, the derivative with respect to 𝑥 is 𝑛𝑎𝑥 raised to the power 𝑛 minus one. That is, we multiply by the exponent of 𝑥 and subtract one from the exponent of 𝑥. We can use this method of differentiation on each of the terms in our expression, differentiating term by term. Beginning with the first term in our expression, our exponent 𝑛 is equal to five. And to differentiate this with respect to 𝑥, we multiply by the exponent five, so we have two times five times 𝑥 raised to the power five minus one, subtracting one from the exponent. That is 10𝑥 to the power four.
In our second term 20𝑥 raised to the fourth power, we have an exponent of 𝑛 equal to four. Differentiating this, we multiply by four and subtract one from the power, which gives us 80𝑥 cubed. In our third term, our exponent 𝑛 is equal to two so that our derivative is negative 57 times the exponent two times 𝑥 raised to the power two minus one, which is negative 114 times 𝑥 raised to the power of one, and 𝑥 raised to the power one is simply 𝑥. In our final term, we have negative 126𝑥. That’s 𝑥 raised to the power one.
And differentiating this with respect to 𝑥, we have negative 126 times one times 𝑥 raised to the power one minus one. That is negative 126 times 𝑥 raised to the power zero. And we know that anything to the power zero is equal to one so that our final term is negative 126. Our first derivative of 𝑦 with respect to 𝑥 is therefore 10𝑥 raised the power four plus 80𝑥 cubed minus 114𝑥 minus 126. To find our second derivative d squared 𝑦 by d𝑥 squared, we differentiate the first derivative d𝑦 by d𝑥, which we’ve just found, with respect to 𝑥. And we can do this using exactly the same method, so term by term.
In our first term, we have 10 times the exponent four times 𝑥 raised to the power four minus one, which is 40𝑥 cubed. And in our second term, we have 80 times the exponent three times 𝑥 raised to the power three minus one, which is 240𝑥 squared. And then our third term, we have negative 114 times the exponent one times 𝑥 raised to the power one minus one, which is negative 114 since 𝑥 raised to the power zero is one. Our fourth term is the constant negative 126. And since a constant doesn’t change with respect to 𝑥, its derivative is equal to zero.
Our second derivative, therefore, is 40𝑥 raised to the power three plus 240𝑥 squared minus 114. And we’ve highlighted this since we’ll come back to it later. Now, for our third derivative with respect to 𝑥, we use the power rule again, differentiating the second derivative. Differentiating 40𝑥 cubed, we have 40 times three times 𝑥 raised to the three minus one. Differentiating 240𝑥 squared, we have 240 times the exponent two times 𝑥 raised to the power two minus one. And differentiating the constant negative 114 gives us zero. Our third derivative is therefore 120𝑥 squared plus 480 times 𝑥.
And our final derivative d to the fourth of 𝑦 by d𝑥 to the fourth is 120 times the exponent two times 𝑥 raised to the power two minus one plus 480 times one times 𝑥 raised to the power one minus one. That is 240𝑥 plus 480. Now the question asks for the second derivative d squared 𝑦 by d𝑥 squared at the fourth derivative d to the fourth of 𝑦 by d𝑥 to the fourth equals zero. And this means we have to solve for 𝑥 the equation 240𝑥 plus 480 is equal to zero. The first thing we can do to solve this is subtract 480 from both sides. This gives us 240𝑥 is equal to negative 480. And now if we divide both sides by 240, these cancel on the left-hand side, and we find 𝑥 is equal to negative two. So 𝑥 is negative two is the value for which the fourth derivative of 𝑦 with respect to 𝑥 is equal to zero.
And now our final step is to substitute this back into our second derivative of 𝑦. And this gives us 40 times negative two cubed plus 240 times negative two squared minus 114. That is 40 times negative eight plus 240 times four minus 114. And that’s negative 320 plus 960 minus 114, which evaluates to 526 so that the value of d squared 𝑦 by d𝑥 squared at d to the fourth of 𝑦 by d𝑥 to the fourth is equal to zero given 𝑦 is equal to two 𝑥 raised to the fifth power plus 20𝑥 raised to the fourth power minus 57𝑥 squared minus 126𝑥 is 526.
