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

Find the first derivative of π(π₯) = (9π₯Β² β π₯ β 7)(7π₯Β² β 8π₯ β 7) at π₯ = β1.

04:21

### Video Transcript

Find the first derivative of π of π₯ equals nine π₯ squared minus π₯ minus seven multiplied by seven π₯ squared minus eight π₯ minus seven at π₯ equals negative one.

Weβve been asked to find the first derivative of this function at a given value of π₯. To do this, we need to differentiate the function and then substitute π₯ equals negative one. The function weβ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 π’ and π£, the derivative of their product π’π£ is equal to π’ times dπ£ by dπ₯ plus π£ times dπ’ by dπ₯. 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 π’ equal the first polynomial and π£ equal the second. We need to find the derivative of each function separately. And as theyβre both polynomials, we need to recall the power rule of differentiation. This states that for real constants π and π, the derivative with respect to π₯ of π multiplied by π₯ to the πth power is πππ₯ to the π minus first power. In other words, we multiply by the exponent and then reduce the exponent by one. Applying this rule to function π’ gives dπ’ by dπ₯ equals 18π₯ minus one. Remember, the derivative of a constant with respect to π₯ is simply zero. Applying the same rule to π£ gives dπ£ by dπ₯ equals 14π₯ minus eight.

Next, we substitute each of these expressions into the product rule, giving π prime of π₯ equals nine π₯ squared minus π₯ minus seven multiplied by 14π₯ minus eight plus seven π₯ squared minus eight π₯ minus seven multiplied by 18π₯ minus one. Distributing the first set of parentheses gives 126π₯ cubed minus 72π₯ squared minus 14π₯ squared plus eight π₯ minus 98π₯ plus 56. And then distributing the second set of parentheses gives 126π₯ cubed minus seven π₯ squared minus 144π₯ squared plus eight π₯ minus 126π₯ plus seven.

Next, we need to simplify by grouping the like terms in this expression. This gives π prime of π₯ equals 252π₯ cubed minus 237π₯ squared minus 208π₯ plus 63. So, weβve found an expression for the first derivative of the function π of π₯. We now need to evaluate this derivative when π₯ equals negative one. Substituting π₯ 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. Thatβs negative 252 minus 237 plus 208 plus 63, which is negative 218.

So, by using the product rule to differentiate the given function, weβve found that the first derivative of π of π₯ at π₯ equals negative one is negative 218. Note that it would also have been possible to substitute π₯ equals negative one into the unsimplified expression for the derivative and evaluate at this point.