The second function.

As a second example of a function, there's a function called name.

So fn prime of x times the second function, g n of y, plus the first function, fn of x, times the derivative of the second function.

If that's the case, then f prime of x is equal to the derivative of the first function times a second function plus the first function times the derivative of the second function.

So the derivative of x is just 1, times the second function, times e to the x, plus the first function, x, times the derivative of the second function.

Then that equals the derivative of the first times the second function, plus the first function times the derivative of the second.

Now you add that to the derivative of the second function times the first function.

So plus f1 of x, that's just the first function, times the derivative of the second function.

Now the derivative of the second function, it's going to be this function with respect to y.

Let's take the second derivative of our combined area function.

So we're going to use the calculus function here, so second, calculus.

Well the chain rule just told us that this is just the same thing as, let's say, the derivative of the first function f prime of x times the second function g of x plus now the first function f of x times the derivative of the second function.

So if we have u times v, if we take the derivative with respect to t of that, that's equal to the derivative of the first times the second function plus the first function times the derivative of the second function.

And then to that we will add the derivative of this second function.

So say some constant, c1, times my first function, f of t, plus some constant, c2, times my second function, g of t.

Well, the second derivative is just going to be the slope of this function.

The derivative of that second function is 40x minus 21x to the minus eighth.

This is also non linear, because I multiplied the function times its second derivative.

Well, we have our second quadrant intersection as a function of our first quadrant x.

That's because it's a quadratic function we have an x squared term, a second degree term, here.

The second derivative of y minus 3 times the first derivative y minus 4 times the function.

That the chain rule, with respect to one of the variables, but the second variable in the function is also a function of x, the chain rule is this.

So the second derivative of my function f, let's see, 3 times 12 is 36x squared minus 24x.

Let's do another problem where we graph a function based on the properties of its derivatives and second derivative.

The derivative of the second function with respect to y is just g prime of y times dy dx.

So draw back into your brain and think, is there some function that when I take its first and second derivatives, and third and fourth derivatives, it essentially becomes the same function?

Form is function. Form is function.

3.7.2.1 In the second sentence, replace reproductive ability and capacity with sexual function and fertility and delete as separate issues .

The second statement is saying look, if I apply f to f inverse, I'm getting the identity function on Y.

Well, that's just going to be f1 prime of x times the second function, well, that's just g1 of y.

So let's say that I have the second derivative of my function y plus 4 times my function y is equal to sine of t minus the unit step function 0 up until 2 pi of t times sine of t minus 2 pi.

So 1 is the derivative of x times the second expression, or you could say the second mini function or whatever, times y, and then plus the first expression x times the derivative of the second expression.

Well, we can take the second derivative of this function, see what the second derivative is at x equals 4, and then see if we're concave upwards or concave downwards.

So, product rule, derivative of the first, v prime times the second expression, e to the minus 2x, plus the first function, or expression, times the derivative of the second.

How fast is y changing as a function of time when x is changing at a function of time at 4 meters per second, and the value of x is equal to 8 meters.

That function and its derivatives and its second derivatives must be something of the form, something times e to the 2x.

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