Question Video: Differentiating Polynomial Functions in the Factorised Form Mathematics

Video Transcript

Find the first derivative of the function 𝑦 equals three 𝑥 to the power of five plus seven multiplied by seven minus three 𝑥 to the power of five.

Here, we have an equation which is the product of two functions. Since this is the case, we can find the derivative of this function by applying the product rule. Remember this rule says that the derivative of the product of 𝑓 and 𝑔 is equal to 𝑓 times the derivative of 𝑔 plus the derivative of 𝑓 times 𝑔. It’s always sensible to write down what we actually know about the expression we’re looking to differentiate. We can say that it’s the product of two functions. Let’s call the first one 𝑓 of 𝑥 three 𝑥 to the power of five plus seven. And we’ll call the second 𝑔 of 𝑥. It’s seven minus three 𝑥 to the power of five. We’re going to need to find the first derivative of each of these functions.

Remember the derivative of a simple algebraic expressions such as 𝑎𝑥 to the power of 𝑛, where 𝑎 and 𝑛 are real constants, is 𝑛 times 𝑎𝑥 to the power of 𝑛 minus one. We multiply by the exponent and then reduce that exponent by one. And what this does actually mean is that the derivative of a constant is zero since a constant — let’s say the number one — is actually one 𝑥 to the power of zero. When we multiply by the exponent, we actually just end up with an answer of zero. This means the derivative of 𝑓 of 𝑥 is five times three 𝑥 to the power of four plus zero which is simply 15𝑥 to the power of four. Similarly, the derivative of 𝑔 of 𝑥 is zero minus five times three 𝑥 to the power of four which is negative 15𝑥 to the power of four.

Let’s substitute what we know into the formula for the product rule. d𝑦 by d𝑥 is equal to three 𝑥 to the power of five plus seven times negative 15𝑥 to the power of four plus 15𝑥 to the power of four times seven minus three 𝑥 to the power of five. We then distribute these parentheses by multiplying each term by the term outside. And we get negative 45𝑥 to the power of nine minus 105𝑥 to the power of four plus 105𝑥 to the power of four minus 45𝑥 to the power of nine. Negative 105𝑥 to the power of four plus 105𝑥 to the power of four is zero. And we can see that the first derivative of our function is negative 90𝑥 to the power of nine. It’s also useful to know that we can apply this rule to find the derivative at a given point.