Chapter 4

Find the derivative with respect to the independent variable. $$ f(x)=\frac{\cos (2 x)}{\tan (4 x)} $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\left(\ln \left(1-x^{2}\right)\right)^{3} $$

Differentiate $$ h(t)=\sqrt{a}(t-a)+a $$ with respect to \(t\). Assume that \(a\) is a positive constant.

Differentiate $$ h(s)=a^{4} s^{2}-a s^{4}+\frac{s^{2}}{a^{4}} $$ with respect to \(s\). Assume that \(a\) is a positive constant.

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 \(x=4\)

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 \(f(x)\) at the point \(x=a\) : $$\lim _{h \rightarrow 0} \frac{(a+h)^{2}-a^{2}}{h}$$ Find \(f(x)\).

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 with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln \sqrt{x^{2}+1} $$

Suppose that \(f(2)=-4, g(2)=3, f^{\prime}(2)=1\), and \(g^{\prime}(2)=-2\). Find $$ (f g)^{\prime}(2) $$

Differentiate $$ V(t)=V_{0}(1+\gamma t) $$ with respect to \(t\). Assume that \(V_{0}\) and \(\gamma\) are positive constants.

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=\left|2 x^{2}-1\right| $$

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

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 \(f(x)\) at the point \(x=a\) : $$\lim _{h \rightarrow 0} \frac{4(a+h)^{3}-4 a^{3}}{h}$$ Find \(f(x)\).

Find the derivative with respect to the independent variable. $$ f(x)=\sin x \cos x $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln \sqrt{2 x^{2}-x} $$

Suppose that \(f(2)=-4, g(2)=3, f^{\prime}(2)=1\), and \(g^{\prime}(2)=-2\). Find $$ \left(f^{2}+g^{2}\right)^{\prime}(2) $$

Differentiate $$ p(T)=\frac{N k T}{V} $$ with respect to \(T\). Assume that \(N, k\), and \(V\) are positive constants.

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.)$$ f(x)=\left\\{\begin{array}{cl} x & \text { for } x \leq 0 \\ x+1 & \text { for } x>0 \end{array}\right. $$

In Problems 36-39, assume that \(f(x)\) and \(g(x)\) are differentiable. Find \(\frac{d}{d x} f(2 x)\)

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 \(x \pm|\Delta x| .\) . \(f(x)=3 x^{2}, x=2 \pm 0.1\)

The following limit represents the derivative of a function \(f(x)\) at the point \(x=a\) : $$\lim _{h \rightarrow 0} \frac{\frac{1}{(2+h)^{2}+1}-\frac{1}{5}}{h}$$ Find \(f\) and \(a\).

Find the derivative with respect to the independent variable. $$ f(x)=\sin (2 x-1) \cos (3 x+1) $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln \frac{x}{x+1} $$

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

Differentiate $$ g(N)=N\left(1-\frac{N}{K}\right) $$ with respect to \(N\). Assume that \(K\) is a positive constant.

In Problems 36-39, assume that \(f(x)\) and \(g(x)\) are differentiable. Find \(\frac{d}{d x}\left(\frac{f(x)}{g(x+1)}\right)\)

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 \(x \pm|\Delta x| .\) \(f(x)=\sqrt{x}, x=10 \pm 0.5\)

The following limit represents the derivative of a function \(f\) at the point \((a, f(a))\) : $$\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{6}+h\right)-\sin \frac{\pi}{6}}{h}$$ Find \(f\) and \(a\).

Find the derivative with respect to the independent variable. $$ f(x)=\tan x \cot x $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln \frac{2 x}{1+x^{2}} $$

Assume that \(f(x)\) is differentiable. Find an expression for the derivative of \(y\) at \(x=1\), assuming that \(f(1)=2\) and \(f^{\prime}(1)=-1\) \(y=3 x^{2} f(x)\)

Differentiate $$ g(N)=r N\left(1-\frac{N}{K}\right) $$ with respect to \(N\). Assume that \(K\) and \(r\) are positive constants.

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ f(x)=\left\\{\begin{array}{cl} x^{2} & \text { for } x \leq-1 \\ 2-x^{2} & \text { for } x>-1 \end{array}\right. $$

In Problems 36-39, assume that \(f(x)\) and \(g(x)\) are differentiable. Find \(\frac{d}{d x} f[g(x)+1] .\)

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 \(x \pm|\Delta x| .\) \(f(x)=e^{x}, x=2 \pm 0.2\)

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

Assume that \(f(x)\) is differentiable. Find an expression for the derivative of \(y\) at \(x=1\), assuming that \(f(1)=2\) and \(f^{\prime}(1)=-1\) \(y=-5 x^{3} f(x)-2 x\)

Differentiate $$ g(N)=r N^{2}\left(1-\frac{N}{K}\right) $$ with respect to \(N\). Assume that \(K\) and \(r\) are positive constants.

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ f(x)=\left\\{\begin{array}{cl} x^{4}+1 & \text { for } x \leq 0 \\ e^{-x} & \text { for } x>0 \end{array}\right. $$

In Problems 36-39, assume that \(f(x)\) and \(g(x)\) are differentiable. Find \(\frac{d}{d x}\left(\frac{f(2 x)}{g(2 x)+2 x}\right)\).

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 \(x \pm|\Delta x| .\) \(f(x)=\sin x, x=-1 \pm 0.05\)

Find the derivative with respect to the independent variable. $$ f(x)=\sec x \cos x $$

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

Assume that \(f(x)\) is differentiable. Find an expression for the derivative of \(y\) at \(x=1\), assuming that \(f(1)=2\) and \(f^{\prime}(1)=-1\) \(y=\frac{x f(x)}{2}\)

Differentiate $$ g(N)=r N(a-N)\left(1-\frac{N}{K}\right) $$ with respect to \(N\). Assume that \(r, a\), and \(K\) are positive constants.

In Problems \(40-46\), find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. y=\left(\sqrt{1-2 x^{2}}+1\right)^{2}

Assume that the measurement of \(x\) is \(a c-\) curate within \(2 \% .\) In each case, determine the error \(\Delta f\) in the calculation of \(f\) and find the percentage error \(100 \frac{\Delta f}{f} .\) The quantities \(f(x)\) and the true value of \(x\) are given. \(f(x)=4 x^{3}, x=1.5\)

Find the derivative with respect to the independent variable. $$ f(x)=\sin x \sec x $$