We now have

d
dx
[x5 + y8] = 5x4 + 8y7y' =
d
dx
[9] = 0.

Rearranging this, we get
8y7y' = ??????????

-------------------------------------
A table of values for f, g, f ', and g' is given.
x f(x) g(x) f '(x) g'(x)
1 3 2 4 6
2 1 8 5 7
3 7 2 7 9
(a) If h(x) = f(g(x)), find h'(3).
h'(3) =

(b) If
H(x) = g(f(x)), find H'(1).
H'(1) =

-------------------------------------

Let f and g be the functions in the table below.
x f(x) g(x) f '(x) g'(x)
1 3 2 4 6
2 1 3 5 7
3 2 1 7 9
(a) If F(x) = f(f(x)), find F '(3).
F '(3) =

(b) If G(x) = g(g(x)), find G'(2).
G'(2) =

--------------------------------------

If h(x) = Squareroot 4 + 3f(x), where f(3) = 4 and f '(3) = 3, find h'(3).
h'(3) =
-----------------------------------------
Write the composite function in the form
f(g(x)). [Identify the inner function
u = g(x and the outer function y = f(u).]
y = e^ 5 Squareroot x
(g(x), f(u)) =
Find the derivative dy/dx.

d/dx (x^5 + y^8) is indeed 5x^4 + 8y^7 y'

Usually this is done in conjunction with implicit derivatives, where you have some equation like

x^5 + y^8 = 7
and you wind up with
y' = -5x^4/8y^7
which you then have to evaluate at some point.

------------------------------

h(x) = f(g(x))
h' = f'(g(x)) * g', so
h'(3) = f'(g(3)) g'(3) = f'(2) g'(3) = 5*9 = 45

b is the same thing
------------------------------

This is just the same, but using different functions

------------------------------

h(x) = √(4 + 3f(x))
h' = 3f'/2√(4+3f)
h'(3) = 3*3/√(4+3*4) = 9/√16 = 9/4

------------------------------
If
f(x) = e^x
g(x) = 5√x, then
y = f(u) = e^u = e^(5√x)

y' = 5/(2√x) * e^5√x

Oh, math problems! Let's see if I can bring some humor into them.

For the first problem:

8y7y' = the combination of y and y' seems suspiciously like a math pun waiting to happen. I should consult my mathematician friends, they always have the best jokes about derivatives.

For the second problem:

(a) If h(x) = f(g(x)), finding h'(3) sounds like a detective movie plot. The case of the missing derivative! Will h'(3) ever be found?

(b) If H(x) = g(f(x)), finding H'(1) feels like trying to untangle headphone wires. One wrong move and you end up with a big mess!

For the third problem:

(a) If F(x) = f(f(x)), finding F '(3) seems like opening a Russian nesting doll of derivatives. How many layers deep will we go?

(b) If G(x) = g(g(x)), finding G'(2) is like playing a game of "double the fun, double the derivative headaches!"

And for the last problem:

Finding h'(3) for h(x) = Square root 4 + 3f(x)... Hmm, I'm drawing a blank here. I'll need my math humor manual for this one.

For the composite function:

f(g(x)) = e^5 Square root x... Is f(g(x)) a new form of math poetry? I can imagine mathematicians reciting it at open mic nights!

As for the derivative dy/dx... well, let's just say it's a derivative of epic proportions. It's like trying to climb Mount Everest in a clown wig!

I hope my attempts at humor brought a smile to your face. Remember, laughter is the best derivative!

To find the derivative of the given composite function, we need to use the chain rule.

The chain rule states that if we have a composite function y = f(u) and u = g(x), then the derivative of y with respect to x is given by dy/dx = dy/du * du/dx.

In this case, the inner function is u = g(x) = √x, and the outer function is y = f(u) = e^(5√x).

Now let's find the derivatives dy/du and du/dx step by step.

1. Derivative of the outer function (dy/du):
Since y = e^(5√x), the derivative of y with respect to u (dy/du) is obtained by applying the chain rule to the exponent. The derivative of e^(5u) with respect to u is simply e^(5u).

Therefore, dy/du = e^(5√x).

2. Derivative of the inner function (du/dx):
The inner function is u = √x. The derivative of √x with respect to x (du/dx) can be found using the power rule. The power rule states that the derivative of x^(n/2) is (n/2)x^(n/2-1).

Applying the power rule, we get du/dx = (1/2)x^(-1/2).

3. Final derivative (dy/dx):
We can use the product rule to find the derivative of y with respect to x (dy/dx) by multiplying dy/du and du/dx.

dy/dx = dy/du * du/dx
= e^(5√x) * (1/2)x^(-1/2).

So, the derivative of the composite function dy/dx is e^(5√x) * (1/2)x^(-1/2).

To find the derivative of a function, you need to use the rules of differentiation, such as the power rule, product rule, and chain rule. Here are the steps to find the derivatives of the given functions:

1. For the function 8y^7y', we can use the product rule since it is a product of two functions, y^7 and y'. Apply the product rule as follows:
dy/dx = y^7 * (d(8)/dx) + 8y' * (d(y^7)/dx)
= 0 + 8y' * 7y^6
= 56y^6y'

2. For the function h(x) = f(g(x)), to find h'(3), we need to differentiate h(x) with respect to x and then substitute x = 3. Use the chain rule to find h'(x) as follows:
h'(x) = f'(g(x)) * g'(x)
h'(3) = f'(g(3)) * g'(3)

To find the values of f'(g(3)) and g'(3), we need to substitute x = 3 in the given table for f(x), g(x), f'(x), and g'(x).

(a) h'(3) = f'(g(3)) * g'(3)
= f'(2) * 9
= 1 * 9
= 9

(b) To find H'(1) for H(x) = g(f(x)), follow the same steps as above:
H'(1) = g'(f(1)) * f'(1)
= g'(3) * 4
= 6 * 4
= 24

3. For the function F(x) = f(f(x)), to find F'(3), we need to differentiate F(x) with respect to x and then substitute x = 3. Use the chain rule as follows:
F'(x) = f'(f(x)) * f'(x)
F'(3) = f'(f(3)) * f'(3)

To find the values of f'(f(3)) and f'(3), we need to substitute x = 3 in the given table for f(x), g(x), f'(x), and g'(x).

(a) F'(3) = f'(f(3)) * f'(3)
= f'(2) * 7
= 4 * 7
= 28

4. For the function G(x) = g(g(x)), to find G'(2), we need to differentiate G(x) with respect to x and then substitute x = 2. Use the chain rule as follows:
G'(x) = g'(g(x)) * g'(x)
G'(2) = g'(g(2)) * g'(2)

To find the values of g'(g(2)) and g'(2), we need to substitute x = 2 in the given table for f(x), g(x), f'(x), and g'(x).

(b) G'(2) = g'(g(2)) * g'(2)
= g'(3) * 7
= 6 * 7
= 42

5. For the function h(x) = √(4 + 3f(x)), to find h'(3), we need to differentiate h(x) with respect to x and then substitute x = 3. Use the chain rule as follows:
h'(x) = (1/2) * (4 + 3f(x))^(-1/2) * 3f'(x)
h'(3) = (1/2) * (4 + 3f(3))^(-1/2) * 3f'(3)

To find the values of f(3) and f'(3), refer to the given information: f(3) = 4 and f'(3) = 3.

h'(3) = (1/2) * (4 + 3(4))^(-1/2) * 3(3)
= (1/2) * (4 + 12)^(-1/2) * 9
= (1/2) * 16^(-1/2) * 9
= (1/2) * (1/4) * 9
= 9/8

6. For the function y = e^(5√x), we can identify the inner function as u = √x and the outer function as y = e^u. To find dy/dx, we apply the chain rule.

dy/dx = dy/du * du/dx

To find dy/du and du/dx, we need to differentiate y with respect to u and differentiate u with respect to x.

dy/du = e^u
du/dx = (1/2) * x^(-1/2) = (1/2√x)

dy/dx = (e^(√x)) * (1/2√x)

So, the derivative dy/dx of y = e^(5√x) is (e^(√x)) * (1/2√x).