### Video Transcript

Determine the equation of the tangent to the curve π¦ is equal to π₯ squared minus eight π₯ times π₯ cubed plus three at the point negative one, 18.

The question wants us to find the equation of the tangent to our curve at the point negative one, 18. We recall the tangent line to our curve at the point negative one, 18 will be a straight line with the same slope as our curve at this point. And it will also pass through this point. And we know the general equation for a straight line. A line of slope π which passes through the point π₯ one, π¦ one will have equation π¦ minus π¦ one is equal to π times π₯ minus π₯ one.

Since our tangent line passes through the point negative one, 18, we can set π₯ one to be negative one and π¦ one to be 18. So the equation of our straight line becomes π¦ minus 18 is equal to π times π₯ minus negative one. The only thing we need to find is our slope π. And remember, the slope of our tangent line will be the same as the slope of the curve at the point negative one, 18. And we know how to find the slope of a curve. Weβll do this by using the derivative. So we want to find an expression for dπ¦ by dπ₯. Thatβs the derivative of our curve π₯ squared minus eight π₯ times π₯ cubed plus three with respect to π₯.

And thereβs a few different ways we can evaluate this derivative. We could notice that this is the product of two functions and use the product rule. However, we could also just multiply these factors together by using the FOIL method. We multiply our first terms together, and we get π₯ squared times π₯ to the third power, which is π₯ to the fifth power. Next, we multiply our outer terms together. This gives us π₯ squared multiplied by three, which is three π₯ squared. Next, we multiply our inner terms together. This gives us negative eight π₯ times π₯ cubed, which is equal to negative eight π₯ to the fourth power. Finally, we multiply the last two terms together. This gives us negative eight π₯ multiplied by three, which is equal to negative 24π₯.

Now, we can see we just need to evaluate the derivative of a polynomial. We can do this by using the power rule for differentiation. For each term, we multiply by the exponent of π₯ and then reduce this exponent by one. This gives us five π₯ to the fourth power plus six π₯ minus 32π₯ cubed minus 24. So this expression tells us the slope of our curve at π₯. We want to know the slope of our curve at the point negative one, 18. Thatβs when π₯ is equal to negative one.

So weβll substitute π₯ is equal to negative one into the equation for the slope of our curve. This gives us the slope of our curve when π₯ is equal to negative one is given by five times negative one to the fourth power plus six times negative one minus 32 times negative one cubed minus 24. And now, we can just calculate this expression. Simplifying each term, we get five minus six plus 32 minus 24 which we can calculate to give us seven. So this is the slope of our curve when π₯ is equal to negative one.

And since this is the same as the slope of our tangent line, we have π is equal to seven. Substituting in π is equal to seven into our equation for the straight line, we get π¦ minus 18 is equal to seven times π₯ minus negative one. We can now simplify this expression. π₯ minus negative one is the same as π₯ plus one. Next, weβll distribute our factor of seven over our parentheses. This gives us π¦ minus 18 is equal to seven π₯ plus seven. Now, weβll subtract seven π₯ plus seven from both sides of this equation. This gives us π¦ minus 18 minus seven π₯ minus seven is equal to zero. Finally, we can simplify this expression, since negative 18 minus seven is equal to negative 25.

Therefore, weβve shown that the tangent to the curve π¦ is equal to π₯ squared minus eight times π₯ cubed plus three at the point negative one, 18 has the equation π¦ minus seven π₯ minus 25 is equal to zero.