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.
