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# Prove a property of the derivative if arctangent and the logarithm obey a given relation

If prove that Proof. First, we consider the derivatives of the left and right side of the given equation. (Treating as a function of and remembering to use the chain rule.) So, for the derivative on the left, we have On the right we have, Now, using the given equation we have # Determine the derivative of a function g in terms of a function f

Determine the derivative of the function in terms of the function for the following definitions of :

1. ;
2. ;
3. ;
4. .

1. We use the chain rule to compute, 2. We use the chain rule to compute, 3. We use the chain rule to compute, 4. Finally we use part (c) and the chain rule to compute, # Determine the value of g(x) = xf(x^2) and its derivatives given values of f(x)

Given a function and the following table of values for and its derivatives: Compute the value of , and for .

First, by the chain rule we have Using these formulas we compute the table, # Compute values of a composite function given values of the components

Given functions and with the following values Let and and construct a table as above for the functions and .

First, by the chain rule we have Using these formulas we compute the table, # Compute the derivatives of two given functions

Consider the functions Find thd derivatives and .

First, we compute the derivative of , where we multiplied the numerator and denominator by to get the second line.

Next, using the chain rule we know So, then using the quotient rule and chain rule we can evaluate the derivative of , Then, plugging in the definition of and the expression for which we already computed, we have, # Compute the derivative of the given function

Compute the derivative of the function where .

We compute this directly, (we can use this exercise to quickly get the formula for the derivative of a product of three differentiable functions, although it’s not really necessary since it is still a direct application of the product rule for derivatives) # Compute the derivative of the given function

Compute the derivative of the function To help our computation, let’s define . Then, Then, using this and the chain rule, we have (This is the form the answer in the back of the book gives, so we’ll leave it in this form. If you want, you can plug back in , but I don’t think it simplifies to anything nice.)

# Compute the derivative of the given function

Compute the derivative of the function First, we simplify the expression for , Now, we take the derivative. (Of course, you could take the derivative directly from the given form of , I just find this form somewhat easier, but it was not so obvious we could simplify this way.) # Compute the derivative of the given function

Compute the derivative of the function Using the chain rule we compute, # Compute the derivative of the given function

Compute the derivative of the function Using the chain rule we compute, 