### Video Transcript

A curve passes through zero, one and the tangent at its point ๐ฅ, ๐ฆ has slope six ๐ฅ times the square root of eight ๐ฅ squared plus one. What is the equation of the curve?

The key to answering this question is to spot that weโve been given information about the slope of the tangent at a given point. We remember that we can find a general equation for the slope of a tangent to the curve by differentiating the equation of the curve. In other words, in this case, d๐ฆ by d๐ฅ is equal to six ๐ฅ times the square root of eight ๐ฅ squared plus one. Or we can write this alternatively as six ๐ฅ times eight ๐ฅ squared plus one to the power of one-half.

Now, by recalling the fact that integration and differentiation are the reverse processes of one another, we find that we can find an expression for ๐ฆ by integrating our expression for d๐ฆ by d๐ฅ with respect to ๐ฅ. Now, essentially, this will give us a general equation. Weโll need to use the fact that the curve passes through zero, one to find the particular equation of our curve. So, weโll integrate six ๐ฅ times eight ๐ฅ squared plus one to the power of one-half with respect to ๐ฅ.

Now, you might be wondering how on Earth weโre going to integrate this function. Well, we have the product of two functions, one of which is a composite function. The key here is to spot that the derivative of part of our composite function is equal to a scalar multiple of another part of our function. In other words, the derivative of eight ๐ฅ squared plus one is some multiple of six ๐ฅ. This is a good indication to us that weโre going to use integration by substitution. Weโll make the substitution ๐ข is equal to eight ๐ฅ squared plus one. Itโs the inner part of our composite function.

Next, we differentiate ๐ข with respect to ๐ฅ, remembering that to differentiate a power term, we multiply the entire term by the exponent then reduce the exponent by one. So we get two times eight ๐ฅ to the power of one or two times eight ๐ฅ which is 16๐ฅ. Of course, the derivative of one is zero. So, d๐ข by d๐ฅ is 16๐ฅ. Now, d๐ข by d๐ฅ isnโt a fraction. But in this process, we treat it a little like one. And we say that this is equivalent to a sixteenth d๐ข equals ๐ฅ d๐ฅ. Now, the whole point of doing this is we can now replace eight ๐ฅ squared plus one with ๐ข. And we can replace ๐ฅ d๐ฅ with a sixteenth d๐ข. So this might look like the integral of six times ๐ข to the power of one-half times a sixteenth d๐ข.

But of course, we can rewrite this a little bit more nicely as six over 16 times ๐ข to the power of one-half d๐ข. And then, weโll take out that constant factor of six over 16 or three over eight. And so ๐ฆ is equal to three-eighths times the indefinite integral of ๐ข to the power of half d๐ข. And here itโs worth recalling that to integrate a term of the form ๐ฅ to the ๐th power where ๐ isnโt equal to negative one, we add one to the power and then divide by that new value. So, the integral of ๐ข to the power of one-half is ๐ข to the power of three over two divided by three over two. And of course, we need that constant of integration ๐ถ.

We know that dividing by a fraction is the same as multiplying by the reciprocal of that fraction. And so, ๐ข to the power of three over two divided by three over two is the same as two-thirds ๐ข to the power of three over two. Now, remember, we want an equation for the curve, so that will be ๐ฆ in terms of ๐ฅ. So we go back to our substitution ๐ข is equal to eight ๐ฅ squared plus one. And so we have ๐ฆ equals three-eighths times two-thirds of eight ๐ฅ squared plus one to the power of three over two plus ๐ถ.

Now, really, what we want to do is find the value of ๐ถ. So weโre going to go back to the very first bit of information about the curve and the fact that it passes through zero, one. In other words, when ๐ฅ is equal to zero, ๐ฆ is equal to one. And so, we substitute these values in. One equals three-eighths times two-thirds eight times zero squared plus one to the power of three over two plus ๐ถ. Now, eight times zero squared plus one is just one, and one to the power of three over two is still one. So we get one equals three-eighths times two-thirds plus ๐ถ.

Letโs solve for ๐ถ by dividing through by three-eighths. So eight-thirds is equal to two-thirds plus ๐ถ. And if we subtract two-thirds from both sides, we see ๐ถ is equal to six-thirds which is simply two. Replacing ๐ถ with two, and we do have an equation of the curve, but it doesnโt look very nice. So weโre going to distribute three-eighths across our parentheses. Three-eighths times two-thirds is one-quarter, and three-eighths times two is three-quarters. And so the equation of our curve is ๐ฆ equals a quarter times eight ๐ฅ squared plus one to the power of three over two plus three-quarters.