Find the first derivative of 𝑓 of
𝑥 equals nine 𝑥 squared minus 𝑥 minus seven multiplied by seven 𝑥 squared minus
eight 𝑥 minus seven at 𝑥 equals negative one.
We’ve been asked to find the first
derivative of this function at a given value of 𝑥. To do this, we need to
differentiate the function and then substitute 𝑥 equals negative one. The function we’ve been given is
the product of two polynomials. We could distribute the parentheses
to give a single polynomial function and then differentiate. But we can also approach this
problem by applying the product rule of differentiation. This states that for two
differentiable functions 𝑢 and 𝑣, the derivative of their product 𝑢𝑣 is equal to
𝑢 times d𝑣 by d𝑥 plus 𝑣 times d𝑢 by d𝑥. In other words, we multiply each
function by the derivative of the other and add these expressions together.
For the given function then, we can
let 𝑢 equal the first polynomial and 𝑣 equal the second. We need to find the derivative of
each function separately. And as they’re both polynomials, we
need to recall the power rule of differentiation. This states that for real constants
𝑎 and 𝑛, the derivative with respect to 𝑥 of 𝑎 multiplied by 𝑥 to the 𝑛th
power is 𝑎𝑛𝑥 to the 𝑛 minus first power. In other words, we multiply by the
exponent and then reduce the exponent by one. Applying this rule to function 𝑢
gives d𝑢 by d𝑥 equals 18𝑥 minus one. Remember, the derivative of a
constant with respect to 𝑥 is simply zero. Applying the same rule to 𝑣 gives
d𝑣 by d𝑥 equals 14𝑥 minus eight.
Next, we substitute each of these
expressions into the product rule, giving 𝑓 prime of 𝑥 equals nine 𝑥 squared
minus 𝑥 minus seven multiplied by 14𝑥 minus eight plus seven 𝑥 squared minus
eight 𝑥 minus seven multiplied by 18𝑥 minus one. Distributing the first set of
parentheses gives 126𝑥 cubed minus 72𝑥 squared minus 14𝑥 squared plus eight 𝑥
minus 98𝑥 plus 56. And then distributing the second
set of parentheses gives 126𝑥 cubed minus seven 𝑥 squared minus 144𝑥 squared plus
eight 𝑥 minus 126𝑥 plus seven.
Next, we need to simplify by
grouping the like terms in this expression. This gives 𝑓 prime of 𝑥 equals
252𝑥 cubed minus 237𝑥 squared minus 208𝑥 plus 63. So, we’ve found an expression for
the first derivative of the function 𝑓 of 𝑥. We now need to evaluate this
derivative when 𝑥 equals negative one. Substituting 𝑥 equals negative one
gives 252 multiplied by negative one cubed minus 237 multiplied by negative one
squared minus 208 multiplied by negative one plus 63. That’s negative 252 minus 237 plus
208 plus 63, which is negative 218.
So, by using the product rule to
differentiate the given function, we’ve found that the first derivative of 𝑓 of 𝑥
at 𝑥 equals negative one is negative 218. Note that it would also have been
possible to substitute 𝑥 equals negative one into the unsimplified expression for
the derivative and evaluate at this point.