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Chapter 5: Problem 50

Find the derivative of the function. \(y=\ln \left(t^{2}+4\right)-\frac{1}{2} \arctan \frac{t}{2}\)

Short Answer

Expert verified
The derivative of the given function is \( \frac{7t}{4*(t^{2}+4)} \)

Step by step solution

01

Identify and differentiate the first part

The first part of the function is \(y = \ln (t^2+4)\). The derivative of the natural log function is \(1/x\), and here \(x\) is \(t^2+4\). So using the chain rule, which states that the derivative of \(h(g(x))\) is \(h'(g(x)) \cdot g'(x)\), the derivative of the first part would be \((1/(t^2+4)) \cdot (2t) = 2t/(t^2+4)\).
02

Identify and differentiate the second part

The second part of the function is \(-1/2*\arctan(t/2)\). Here we use the rule that the derivative of \(arctan(u)\) is \(1/(1+u^2)\). Applying this rule, along with the chain rule as in Step 1, we find that the derivative of the second part is \(-1/2 * (1/(1+(t/2)^2)) * (1/2) = -t/(4*(t^2+4))\).
03

Combine the results

By the rule that the derivative of a sum of functions is the sum of the derivatives, we add the derivatives from Step 1 and Step 2: \(2t/(t^2+4) - t/(4*(t^2+4)) = (8t - t) / (4*(t^2 + 4)) = 7t / (4*(t^2+4)).

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Most popular questions from this chapter

Chemical Reactions Chemicals \(A\) and \(B\) combine in a 3 -to- 1 ratio to form a compound. The amount of compound \(x\) being produced at any time \(t\) is proportional to the unchanged amounts of \(A\) and \(B\) remaining in the solution. So, when 3 kilograms of \(A\) is mixed with 2 kilograms of \(B\) , you have $$\frac{d x}{d t}=k\left(3-\frac{3 x}{4}\right)\left(2-\frac{x}{4}\right)=\frac{3 k}{16}\left(x^{2}-12 x+32\right)$$ One kilogram of the compound is formed after 10 minutes. Find the amount formed after 20 minutes by solving the equation $$\int \frac{3 k}{16} d t=\int \frac{d x}{x^{2}-12 x+32}$$

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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. \(f(x)=\arccos x, \quad a=0\)
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