Let r(x)= f(g(h(x))), where h(1)=4, g(4)=5, h'(1)=3 , g'(4)=5 and f'(5)=7. find r'(1).
r = f(g(h(x))
r'(x) = f'(g(h(x))) g'(h(x)) h'(x)
r'(1) = f'(g(h(1))) g'(h(1)) h'(1)
= f'(g(4)) g'(4) (3)
= f'(5) (5)(3)
= (7)(5)(3)
= 105
Well, I'm no math whiz, but I'll give it a clown's best shot!
Let's break it down step by step. We know that h(1) is 4, so h(1) being a lonely loner number is now all snuggled up with 4.
Then, we have g(4) being 5, which means g(4) is looking for a hand to hold and eventually found 5 as their special someone.
As for h'(1) being 3, well, h'(1) is a bit of a speed demon, going at 3 units per whatever-it-is-they're-doing. Zoom zoom!
And g'(4) being 5 means that g'(4) is even faster than h'(1), sprinting at a whopping 5 units per whatever-it-is-they're-doing! They must have had some clownfuel!
Now, let's not forget f'(5) being 7. Oh boy, oh boy! f'(5) is like a little rocket shooting up at 7 units per whatever-it-is-they're-doing! Up, up, and away!
Alright, enough clowning around. To find r'(1), we need to use the chain rule. The chain rule says that if you're wrapping one function inside another function inside another function, you need to multiply all the derivatives together.
So, r'(1) = f'(g(h(1))) * g'(h(1)) * h'(1).
Plugging in the values we know, we get:
r'(1) = f'(g(4)) * g'(4) * h'(1).
r'(1) = f'(5) * g'(4) * h'(1).
r'(1) = 7 * 5 * 3.
And finally, after some multiplication, we get:
r'(1) = 105.
Ta-da! The mighty result is 105. I hope that made you smile, my friend!
To find r'(1), we need to use the chain rule, which states that if h(x), g(x), and f(x) are differentiable functions, then the derivative of r(x) = f(g(h(x))) is given by:
r'(x) = f'(g(h(x))) * g'(h(x)) * h'(x)
Given information:
h(1) = 4
g(4) = 5
h'(1) = 3
g'(4) = 5
f'(5) = 7
To find r'(1), we substitute the given values into the chain rule formula:
r'(1) = f'(g(h(1))) * g'(h(1)) * h'(1)
Substituting the values:
r'(1) = f'(g(4)) * g'(h(1)) * h'(1)
r'(1) = f'(5) * g'(4) * h'(1)
Now, substitute the given values:
r'(1) = 7 * 5 * 3
r'(1) = 105
Therefore, r'(1) = 105.
To find r'(1), we need to differentiate the function r(x) with respect to x and then evaluate it at x=1. To differentiate r(x), we use the chain rule.
The chain rule states that if we have a composite function y = f(g(h(x))), then its derivative is given by:
dy/dx = f'(g(h(x))) * g'(h(x)) * h'(x)
Here's how we can apply the chain rule to find r'(1):
Step 1: Calculate h'(x):
Given h'(1) = 3, we know that h'(x) = 3.
Step 2: Calculate g'(x):
Given g'(4) = 5, we know that g'(x) = 5.
Step 3: Calculate f'(x):
Given f'(5) = 7, we know that f'(x) = 7.
Step 4: Find r'(x):
Using the chain rule, we have:
r'(x) = f'(g(h(x))) * g'(h(x)) * h'(x)
Substituting the known values:
r'(x) = 7 * 5 * 3 = 105
Step 5: Evaluate r'(x) at x=1:
r'(1) = 105
Therefore, r'(1) = 105.