Question Video: Finding the First Derivative of Polynomial Functions Using Product Rule at a Point Mathematics • Higher Education

Video Transcript

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.