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Chapter 5: Problem 44

Find the derivative of the function. \(f(x)=\arctan \sqrt{x}\)

Short Answer

Expert verified
The derivative of the function \(f(x)=\arctan \sqrt{x}\) is \(f'(x) = \frac{1}{2(x+1)x^{1/2}}\)

Step by step solution

01

Identify the expression for differentiation

Our main task is to differentiate the function \(f(x) = \arctan \sqrt{x}\) this is a composition of functions. First, we have \(\sqrt{x}\) and then \(\arctan \(x)\). We'll begin by differentiating the outermost function using the chain rule.
02

Apply the chain rule

The chain rule states that the derivative of a composite function is given by the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. By applying chain rule, we have \(f'(x) = \frac{1}{1 + (\sqrt{x})^2} * \frac{1}{2}x^{-1/2}\) as the derivative of our original function.
03

Simplify the result

We can simplify our derived function to obtain \(f'(x) = \frac{1}{2(x+1)x^{1/2}}\) by multiplying out our previous result.

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Most popular questions from this chapter

Find the derivative of the function. \(g(x)=3 \arccos \frac{x}{2}\)

Use a computer algebra system to find the linear approximation \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and the quadratic approximation \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) of the function \(f\) at \(x=a\) . Sketch the graph of the function and its linear and quadratic approximations. \(f(x)=\arctan x, \quad a=0\)

(a) Graph the function \(f(x)=\arccos x+\arcsin x\) on the interval \([-1,1] .\) (b) Describe the graph of \(f\) . (c) Verify the result of part (b) analytically.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is an even function, then \(f^{-1}\) exists.

Proof Prove that $$\tanh ^{-1} x=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad-1
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