Chapter 3

Find the derivative of \(y\) with respect to the given independent variable. $$y=5^{\sqrt{s}}$$

In Exercises \(51-70,\) find \(d y / d t\). $$y=3 t\left(2 t^{2}-5\right)^{4}$$

Find the value of \(a\) that makes the following function differentiable for all \(x\) -values.$$g(x)=\left\\{\begin{array}{ll} a x, & \text { if } x<0 \\ x^{2}-3 x, & \text { if } x \geq 0 \end{array}\right.$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=2^{\left(s^{2}\right)}$$

Use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval \(I .\) Perform the following steps: a. Plot the function \(f\) over \(I\) b. Find the linearization \(L\) of the function at the point \(a\). c. Plot \(f\) and \(L\) together on a single graph. d. Plot the absolute error \(|f(x)-L(x)|\) over \(I\) and find its maximum value. e. From your graph in part (d), estimate as large a \(\delta>0\) as you can, satisfying $$|x-a|<\delta \Rightarrow|f(x)-L(x)|<\epsilon$$ for \(\epsilon=0.5,0.1,\) and \(0.01 .\) Then check graphically to see if your \(\delta\) -estimate holds true. $$f(x)=\frac{x-1}{4 x^{2}+1},\left[-\frac{3}{4}, 1\right], \quad a=\frac{1}{2}$$

In Exercises \(51-70,\) find \(d y / d t\). $$y=\sqrt{3 t+\sqrt{2+\sqrt{1-t}}}$$

Find the values of \(a\) and \(b\) that make the following function differentiable for -values.$$f(x)=\left\\{\begin{array}{ll}a x+b, & x>-1 \\\b x^{2}-3, & x \leq-1 \end{array}\right.$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=x^{\pi}$$

Use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval \(I .\) Perform the following steps: a. Plot the function \(f\) over \(I\) b. Find the linearization \(L\) of the function at the point \(a\). c. Plot \(f\) and \(L\) together on a single graph. d. Plot the absolute error \(|f(x)-L(x)|\) over \(I\) and find its maximum value. e. From your graph in part (d), estimate as large a \(\delta>0\) as you can, satisfying $$|x-a|<\delta \Rightarrow|f(x)-L(x)|<\epsilon$$ for \(\epsilon=0.5,0.1,\) and \(0.01 .\) Then check graphically to see if your \(\delta\) -estimate holds true. $$f(x)=x^{2 / 3}(x-2), \quad[-2,3], \quad a=2$$

Find \(y^{\prime \prime}\) in Exercises \(71-78\). $$y=\left(1+\frac{1}{x}\right)^{3}$$

The general polynomial of degree \(n\) has the form $$ P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0}$$ where \(a_{n} \neq 0 .\) Find \(P^{\prime}(x)\).

Find the derivative of \(y\) with respect to the given independent variable. $$y=t^{1-e}$$

The reaction of the body to a dose of medicine can sometimes be represented by an equation of the form $$R=M^{2}\left(\frac{C}{2}-\frac{M}{3}\right)$$ where \(C\) is a positive constant and \(M\) is the amount of medicine absorbed in the blood. If the reaction is a change in blood pressure, \(R\) is measured in millimeters of mercury. If the reaction is a change in temperature, \(R\) is measured in degrees, and so on. Find \(d R / d M .\) This derivative, as a function of \(M,\) is called the sensitivity of the body to the medicine. In Section \(4.5,\) we will see how to find the amount of medicine to which the body is most sensitive.

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{2} 5 \theta$$

Use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval \(I .\) Perform the following steps: a. Plot the function \(f\) over \(I\) b. Find the linearization \(L\) of the function at the point \(a\). c. Plot \(f\) and \(L\) together on a single graph. d. Plot the absolute error \(|f(x)-L(x)|\) over \(I\) and find its maximum value. e. From your graph in part (d), estimate as large a \(\delta>0\) as you can, satisfying $$|x-a|<\delta \Rightarrow|f(x)-L(x)|<\epsilon$$ for \(\epsilon=0.5,0.1,\) and \(0.01 .\) Then check graphically to see if your \(\delta\) -estimate holds true. $$f(x)=x 2^{x}, \quad[0,2], \quad a=1$$

Suppose that the function \(v\) in the Derivative Product Rule has a constant value \(c .\) What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule?

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{3}(1+\theta \ln 3)$$

Use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval \(I .\) Perform the following steps: a. Plot the function \(f\) over \(I\) b. Find the linearization \(L\) of the function at the point \(a\). c. Plot \(f\) and \(L\) together on a single graph. d. Plot the absolute error \(|f(x)-L(x)|\) over \(I\) and find its maximum value. e. From your graph in part (d), estimate as large a \(\delta>0\) as you can, satisfying $$|x-a|<\delta \Rightarrow|f(x)-L(x)|<\epsilon$$ for \(\epsilon=0.5,0.1,\) and \(0.01 .\) Then check graphically to see if your \(\delta\) -estimate holds true. $$f(x)=\sqrt{x} \sin ^{-1} x,[0,1], \quad a=\frac{1}{2}$$

Find \(y^{\prime \prime}\) in Exercises \(71-78\). $$y=9 \tan \left(\frac{x}{3}\right)$$

The Reciprocal Rule a. The Reciprocal Rule says that at any point where the function \(v(x)\) is differentiable and different from zero, $$\frac{d}{d x}\left(\frac{1}{v}\right)=-\frac{1}{v^{2}} \frac{d v}{d x}$$ Show that the Reciprocal Rule is a special case of the Derivative Quotient Rule. b. Show that the Reciprocal Rule and the Derivative Product Rule together imply the Derivative Quotient Rule.

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{4} x+\log _{4} x^{2}$$

The Derivative Product Rule gives the formula $$\frac{d}{d x}(u v)=u \frac{d v}{d x}+v \frac{d u}{d x}$$ for the derivative of the product \(u v\) of two differentiable functions of \(x\) . a. What is the analogous formula for the derivative of the product uvw of three differentiable functions of \(x ?\) b. What is the formula for the derivative of the product \(u_{1} u_{2} u_{3} u_{4}\) of four differentiable functions of \(x ?\) c. What is the formula for the derivative of a product \(u_{1} u_{2} u_{3} \cdots u_{n}\) of a finite number \(n\) of differentiable functions of \(x ?\)

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{25} e^{x}-\log _{5} \sqrt{x}$$

Find \(y^{\prime \prime}\) in Exercises \(71-78\). $$y=x^{2}\left(x^{3}-1\right)^{5}$$

Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is, $$\frac{d}{d x}\left(x^{-m}\right)=-m x^{m-1}$$ where \(m\) is a positive integer.

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{2} r \cdot \log _{4} r$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{3} r \cdot \log _{9} r$$

Find \(y^{\prime \prime}\) in Exercises \(71-78\). $$y=\sin \left(x^{2} e^{x}\right)$$

One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is $$A(q)=\frac{k m}{q}+c m+\frac{h q}{2}$$ where \(q\) is the quantity you order when things run low (shoes, TVs, brooms, or whatever the item might be): \(k\) is the cost of placing an order (the same, no matter how often you order); \(c\) is the cost of one item (a constant); \(m\) is the number of items sold each week (a constant); and \(h\) is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security). Find \(d A / d q\) and \(d^{2} A / d q^{2}\)

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{3}\left(\left(\frac{x+1}{x-1}\right)^{\ln 3}\right)$$

In Exercises \(79-84,\) find the value of \((f \circ g)^{\prime}\) at the given value of \(x\). $$f(u)=u^{5}+1, u=g(x)=\sqrt{x}, \quad x=1$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{5} \sqrt{\left(\frac{7 x}{3 x+2}\right)^{\ln 5}}$$

In Exercises \(79-84,\) find the value of \((f \circ g)^{\prime}\) at the given value of \(x\). $$f(u)=1-\frac{1}{u}, \quad u=g(x)=\frac{1}{1-x}, \quad x=-1$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=\theta \sin \left(\log _{7} \theta\right)$$

In Exercises \(79-84,\) find the value of \((f \circ g)^{\prime}\) at the given value of \(x\). $$f(u)=\cot \frac{\pi u}{10}, \quad u=g(x)=5 \sqrt{x}, \quad x=1$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{7}\left(\frac{\sin \theta \cos \theta}{e^{\theta} 2^{\theta}}\right)$$

In Exercises \(79-84,\) find the value of \((f \circ g)^{\prime}\) at the given value of \(x\). $$f(u)=u+\frac{1}{\cos ^{2} u}, \quad u=g(x)=\pi x, \quad x=1 / 4$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{5} e^{x}$$

In Exercises \(79-84,\) find the value of \((f \circ g)^{\prime}\) at the given value of \(x\). $$f(u)=\frac{2 u}{u^{2}+1}, u=g(x)=10 x^{2}+x+1, x=0$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{2}\left(\frac{x^{2} e^{2}}{2 \sqrt{x+1}}\right)$$

In Exercises \(79-84,\) find the value of \((f \circ g)^{\prime}\) at the given value of \(x\). $$f(u)=\left(\frac{u-1}{u+1}\right)^{2}, u=g(x)=\frac{1}{x^{2}}-1, x=-1$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=3^{\log _{2} t}$$

Assume that \(f^{\prime}(3)=-1, g^{\prime}(2)=5, g(2)=3,\) and \(y=f(g(x))\) What is \(y^{\prime}\) at \(x=2 ?\)

Find the derivative of \(y\) with respect to the given independent variable. $$y=3 \log _{8}\left(\log _{2} t\right)$$

If \(r=\sin (f(t)), f(0)=\pi / 3,\) and \(f^{\prime}(0)=4,\) then what is \(d r / d t\) at \(t=0 ?\)

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{2}\left(8 t^{\ln 2}\right)$$

Suppose that functions \(f\) and \(g\) and their derivatives with respect to \(x\) have the following values at \(x=2\) and \(x=3\) $$\begin{array}{lcccc} \hline \boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) & \boldsymbol{g}(\boldsymbol{x}) & \boldsymbol{f}^{\prime}(\boldsymbol{x}) & \boldsymbol{g}^{\prime}(\boldsymbol{x}) \\ \hline 2 & 8 & 2 & 1 / 3 & -3 \\ 3 & 3 & -4 & 2 \pi & 5 \\ \hline \end{array}$$ Find the derivatives with respect to \(x\) of the following combinations at the given value of \(x\) a. \(2 f(x), \quad x=2\) b. \(f(x)+g(x), \quad x=3\) c. \(f(x) \cdot g(x), \quad x=3\) d. \(f(x) / g(x), \quad x=2\) e. \(f(g(x)), \quad x=2\) f. \(\sqrt{f(x)}, \quad x=2\) g. \(1 / g^{2}(x), \quad x=3\) h. \(\sqrt{f^{2}(x)+g^{2}(x)}, \quad x=2\)

Find the derivative of \(y\) with respect to the given independent variable. $$y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)$$

Suppose that the functions \(f\) and \(g\) and their derivatives with respect to \(x\) have the following values at \(x=0\) and \(x=1\) $$\begin{array}{lcccc} \hline \boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) & \boldsymbol{g}(\boldsymbol{x}) & \boldsymbol{f}^{\prime}(\boldsymbol{x}) & \boldsymbol{g}^{\prime}(\boldsymbol{x}) \\ \hline 0 & 1 & 1 & 5 & 1 / 3 \\ 1 & 3 & -4 & -1 / 3 & -8 / 3 \\ \hline \end{array}$$ Find the derivatives with respect to \(x\) of the following combinations at the given value of \(x\) a. \(5 f(x)-g(x), \quad x=1\) b. \(f(x) g^{3}(x), \quad x=0\) c. \(\frac{f(x)}{g(x)+1}, \quad x=1\) d. \(f(g(x)), \quad x=0\) e. \(g(f(x)), \quad x=0\) f. \(\left(x^{11}+f(x)\right)^{-2}, \quad x=1\) g. \(f(x+g(x)), \quad x=0\)

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(x+1)^{x}$$