 Let us find the equation of the tangent line of the function f of x equals the square root of one plus x cubed at the point of tangency being two comma three. You can see a picture of the function f illustrated here in yellow and the tangent line that we're gonna compute its equation for in just a moment in green. So how do we compute the tangent line here? The thing to remember of the tangent line is that it has the following formula. Y minus f of a is equal to f prime at a times x minus a where this is the typical point slope form of a line for which the point x and f of here a and f of a are the x and y coordinate. And then f prime of a is then the slope of the tangent line the derivative gives us the slope of the tangent line. So that's what we need to compute. Now notice some of the things we already do know. For example, we see that the a value is two and then f of a is going to be three. We know that because they're because they told it to you right here. But if you had any doubts you could put two into this function right here. Two cubed is eight plus one is nine. The square root of nine is a three. So that is in fact a point on this graph. The derivative we don't yet know what that is. So we need to compute that. So to find f prime of x we're gonna have to take the derivative of the square root of one plus x cubed right here for which when it comes to square roots we're better off thinking of them as a power function. So I'm gonna write this as one plus x cubed raised to the one half power and then we need to take the derivative of this thing. This is where the chain rule is gonna come into play here because we have two functions. We have this inner function one plus x cubed. This is the functions inside of the square root. It's within the scope of this exponent one half. And then the one half as a power function serves as the outer function. So to compute the derivative by the chain rule we have two parts. We have the outer derivative for which by the power rule we're gonna get one half times one plus x cubed to the negative one half power. Notice by the usual power rule the power comes out in front as a coefficient and then you lower the power by one. One half minus one is a negative one half. But we have to also take the derivative of the inner function, the so-called inner derivative. You have to take the derivative of one plus x cubed. And so considering this thing right here the derivative of one plus x cubed the derivative of one is zero the derivative of x cubed is gonna be three x squared. And then the denominator will look like two times the square root of one plus x cubed that we saw previously. This is our derivative. Now we don't actually need the derivative per se. What we need is the derivative evaluated at two. So as we try to compute f prime at two we could have actually cleaned up our derivative like we did, we didn't actually have to. We could have plugged it in back over here if we preferred. But we need to plug in two into this function to see what happens. So we get three times two squared over two times the square root of one plus two cubed. So let's compute this. We're gonna get a two squared, which is a four. Two on the bottom. Like we saw a moment ago, two cubed is eight plus one is nine, right? So we're gonna get the square root of nine which is a three. We get two times three on the bottom. We get three times four on the top. We see that the threes are gonna cancel. Two goes into four, two times. So the slope of the tangent line is gonna equal to. Plugging those back into the formula we had over here we see that the slope of the tangent line turned out to be two. And therefore we have the equation y minus three is equal to two times x minus two. If you distribute the two, you get two x minus four. That's a minus four. In which case then we have to add three to both sides. Add three, add three. In which case then in the end you end up with y equals two x. You're gonna get negative four plus three which is a negative one which then agrees with the formula of the tangent line that we saw right there.