# Calculus/Chain rule

Question
Hi Paul,
I don't quite know how to approach this question.
Let r(x) = f(g(h(x))), where h(1) = 2, g(2) = 3, h'(1) =4, g'(2) = 5, and f'(3)=6.
Find r' (1).
Do I need to use any of the derivatives rules?
Many Thanks,
Sam

Questioner:Sam
Category:Calculus
Private:No
Subject:derivatives

Question:
Hi Paul,
I don't quite know how to approach this question.
Let r(x) = f(g(h(x))), where h(1) = 2, g(2) = 3, h'(1) =4, g'(2) = 5, and f'(3)=6.
Find r' (1).
Do I need to use any of the derivatives rules?
Many Thanks,
Sam
------------------------------------------------------------
Yes -- the chain rule -- that's what handles composition of functions.
Pretend that x, f, g, h, are variables.

dr/dx = df/dg dg/dh dh/dx

Now let x = 1;  dh/dx = h'(1) = 4

Since h(1) = 2, we could get dg/dh at h = 2 = g'(2) = 5  (nice of them to tell us)

Since g(2) = 3, we can get  df/dg at g = 2 =  f'(3) = 6  (even nicer)

OK, finish up.

Calculus

Volunteer

#### Paul Klarreich

##### Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

##### Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

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(See above.)