Chapter 4

Differentiate the functions with respect to the independent variable. $$ f(x)=\ln \left(1-x^{3}\right) $$

Differentiate $$ f(x)=\frac{r+x}{r s^{2}}-r s x+(r+s) x-r s $$

Suppose that the per capita growth rate of a population is \(3 \% ;\) that is, if \(N(t)\) denotes the population size at time \(t\), then $$\frac{1}{N} \frac{d N}{d t}=0.03$$ Suppose also that the population size at time \(t=4\) is equal to 100\. Use a linear approximation to compute the population size at time \(t=4.1\).

Find the equation of the normal line to the curve \(y=-3 x^{2}\) at the point \((-1,-3)\).

Differentiate $$ g(N)=\frac{b N}{(k+N)^{2}} $$ with respect to \(N\). Assume that \(b\) and \(k\) are positive constants.

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=-3 e^{x^{2}+\tan x} $$

Differentiate $$ f(x)=2 a\left(x^{2}-a\right)^{2}+a $$ with respect to \(x\). Assume that \(a\) is a positive constant.

Differentiate the functions with respect to the independent variable. $$ f(x)=(\ln x)^{2} $$

Differentiate $$ f(N)=(b-1) N^{4}-\frac{N^{2}}{b} $$

Suppose that the per capita growth rate of a population is \(2 \% ;\) that is, if \(N(t)\) denotes the population size at time \(t\), then $$\frac{1}{N} \frac{d N}{d t}=0.02$$ Suppose also that the population size at time \(t=2\) is equal to 50. Use a linear approximation to compute the population size at time \(t=2.1\).

Find the equation of the normal line to the curve \(y=4 / x\) at the point \((-1,-4)\).

Differentiate $$ g(N)=\frac{N}{(k+b N)^{3}} $$ with respect to \(N\). Assume that \(b\) and \(k\) are positive constants.

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ g(t)=\frac{\sin (3 t)}{\cos (5 t)} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=2 e^{-x \sec (3 x)} $$

Differentiate $$ f(x)=\frac{3(x-1)^{2}}{2+a} $$ with respect to \(x\). Assume that \(a\) is a positive constant.

Differentiate the functions with respect to the independent variable. $$ f(x)=(\ln x)^{3} $$

Differentiate $$ f(N)=\frac{b N^{2}+N}{K+b} $$

Suppose that the specific growth rate of a plant is \(1 \% ;\) that is, if \(B(t)\) denotes the biomass at time \(t\), then $$\frac{1}{B(t)} \frac{d B}{d t}=0.01$$ Suppose that the biomass at time \(t=1\) is equal to 5 grams. Use a linear approximation to compute the biomass at time \(t=1.1\).

Find the equation of the normal line to the curve \(y=2 x^{2}-1\) at the point \((1,1)\).

Differentiate $$ g(T)=a\left(T_{0}-T\right)^{3}-b $$ with respect to \(T\). Assume that \(a, b\), and \(T_{0}\) are positive constants.

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ g(t)=\frac{\sin (2 t)+1}{\cos (6 t)-1} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=2^{x} $$

Differentiate the functions with respect to the independent variable. $$ f(x)=\left(\ln x^{2}\right)^{2} $$

Differentiate $$ g(t)=a^{3} t-a t^{3} $$

Suppose that a certain plant is grown along a gradient ranging from nitrogen- poor to nitrogen-rich soil. Experimental data show that the average mass per plant grown in a soil with a total nitrogen content of \(1000 \mathrm{mg}\) nitrogen per \(\mathrm{kg}\) of soil is \(2.7 \mathrm{~g}\) and the rate of change of the average mass per plant at this nitrogen level is \(1.05 \times 10^{-3} \mathrm{~g}\) per mg change in total nitrogen per kg soil. Use a linear approximation to predict the average mass per plant grown in a soil with a total nitrogen content of \(1100 \mathrm{mg}\) nitrogen per \(\mathrm{kg}\) of soil.

Find the equation of the normal line to the curve \(y=\sqrt{x-1}\) at the point \((5,2)\).

Suppose that \(f^{\prime}(x)=2 x+1\). Find the following: (a) \(\frac{d}{d x} f\left(x^{2}\right)\) at \(x=-1\) (b) \(\frac{d}{d x} f(\sqrt{x})\) at \(x=4\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\frac{\cot (2 x)}{\tan (4 x)} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=3^{x} $$

Differentiate the functions with respect to the independent variable. $$ f(x)=\left(\ln \left(1-x^{2}\right)\right)^{3} $$

Differentiate $$ h(s)=a^{4} s^{2}-a s^{4}+\frac{s^{2}}{a^{4}} $$

A measurement error in \(x\) affects the accuracy of the value \(f(x) .\) In each case, determine an interval of the form $$[f(x)-\Delta f, f(x)+\Delta f]$$ that reflects the measurement error \(\Delta x .\) In each problem, the quantities given are \(f(x)\) and \(x=\) true value of \(x \pm|\Delta x|\). $$ f(x)=2 x, x=1 \pm 0.1 $$

The following limit represents the derivative of a function \(f\) at the point \((a, f(a))\) : $$ \lim _{h \rightarrow 0} \frac{2(a+h)^{2}-2 a^{2}}{h} $$

Suppose that \(f^{\prime}(x)=\frac{1}{x}\). Find the following: (a) \(\frac{d}{d x} f\left(x^{2}+3\right)\) (b) \(\frac{d}{d x} f(\sqrt{x-1})\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\frac{\sec \left(x^{2}-1\right)}{\csc \left(x^{2}+1\right)} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=2^{x+1} $$

Differentiate the functions with respect to the independent variable. $$ f(x)=\ln \sqrt{x^{2}+1} $$

Differentiate $$ V(t)=V_{0}(1+\gamma t) $$

A measurement error in \(x\) affects the accuracy of the value \(f(x) .\) In each case, determine an interval of the form $$[f(x)-\Delta f, f(x)+\Delta f]$$ that reflects the measurement error \(\Delta x .\) In each problem, the quantities given are \(f(x)\) and \(x=\) true value of \(x \pm|\Delta x|\). $$ f(x)=1-3 x, x=-2 \pm 0.3 $$

The following limit represents the derivative of a function \(f\) at the point \((a, f(a))\) : $$ \lim _{h \rightarrow 0} \frac{4(a+h)^{3}-4 a^{3}}{h} $$

Assume that \(f(x)\) and \(g(x)\) are differentiable. Find \(\frac{d}{d x} \sqrt{f(x)+g(x)}\).

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\sin x \cos x $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=3^{x-1} $$

Differentiate the functions with respect to the independent variable. $$ f(x)=\ln \sqrt{2 x^{2}-x} $$

Differentiate $$ p(T)=\frac{N k T}{V} $$

A measurement error in \(x\) affects the accuracy of the value \(f(x) .\) In each case, determine an interval of the form $$[f(x)-\Delta f, f(x)+\Delta f]$$ that reflects the measurement error \(\Delta x .\) In each problem, the quantities given are \(f(x)\) and \(x=\) true value of \(x \pm|\Delta x|\). $$ f(x)=3 x^{2}, x=2 \pm 0.1 $$

Assume that \(f(x)\) and \(g(x)\) are differentiable. Find \(\frac{d}{d x}\left(\frac{f(x)}{g(x)}+1\right)^{2}\).

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\sin (2 x-1) \cos (3 x+1) $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=5 \sqrt{2 x-1} $$

In Problems \(37-40\), assume that \(f(x)\) is differentiable. Find an expression for the derivative of \(y\) at \(x=1\), assuming that \(f(1)=2\) and \(f^{\prime}(1)=-1\). $$ y=2 x f(x) $$