Chapter 4

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\sqrt{x^{2}+3} $$

Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a calculator. \(\tan (0.01)\)

Find the derivative at the indicated point from the graph of \(y=f(x)\). \(f(x)=(x+2)^{3} ; x=-2\)

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

Differentiate the functions with respect to the independent variable. \(f(x)=e^{4 x^{2}-2 x+1}\)

Use the product rule to find the derivative with respect to the independent variable. \(f(x)=2\left(3 x^{2}-2 x^{3}\right)\left(1-5 x^{2}\right)\)

Differentiate the functions given with respect to the independent variable. $$ f(x)=-1+3 x^{2}-2 x^{4} $$

In the following examples quantities \(x\) and \(y\) are given. Interpret the role of change dy/dx in words. $$ y \text { is the bite strength of a mammal, } x \text { is its body mass. } $$

Find the first and the second derivatives of each function. $$ f(x)=\frac{1}{x^{2}}+x-x^{3} $$

In Problems \(1-8\), find \(\frac{d y}{d x}\) by implicit differentiation. $$ \frac{1}{2 x y}-y^{3}=4 $$

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\sqrt{2 x+7} $$

Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a calculator. \(\sin \left(\frac{\pi}{2}+0.02\right)\)

Find the derivative at the indicated point from the graph of \(y=f(x)\). \(f(x)=\cos x ; x=0\)

Find the derivative with respect to the independent variable. $$ f(x)=\sin (3 x) $$

Differentiate the functions with respect to the independent variable. \(f(x)=e^{7\left(x^{2}+1\right)^{2}}\)

Use the product rule to find the derivative with respect to the independent variable. \(f(x)=\frac{1}{5}\left(x^{2}-1\right)\left(x^{2}+1\right)\)

Differentiate the functions given with respect to the independent variable. $$ g(s)=5 s^{7}+2 s^{3}-5 s $$

In the following examples quantities \(x\) and \(y\) are given. Interpret the role of change dy/dx in words. $$ y \text { is the body mass of a mammal, } x \text { is its age. } $$

Find the first and the second derivatives of each function. $$ f(s)=s^{3 / 2} $$

In Problems \(1-8\), find \(\frac{d y}{d x}\) by implicit differentiation. $$ \frac{x}{y}=\frac{y}{x} $$

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\sqrt{1-x^{3}} $$

Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a calculator. \(\cos \left(\frac{\pi}{4}-0.01\right)\)

Find the derivative with respect to the independent variable. $$ f(x)=\cos (-5 x) $$

Differentiate the functions with respect to the independent variable. \(f(x)=e^{-3\left(x^{3}-1\right)^{4}}\)

Differentiate the functions given with respect to the independent variable. $$ g(s)=3-4 s^{2}-4 s^{3} $$

In the following examples quantities \(x\) and \(y\) are given. Interpret the role of change dy/dx in words. \(y\) is the temperature of the Pacific Ocean at Santa Monica beach, \(x\) is the time of day.

Find the first and the second derivatives of each function. $$ f(x)=\frac{2 x}{x^{2}+1} $$

In Problems \(1-8\), find \(\frac{d y}{d x}\) by implicit differentiation. $$ \frac{x}{x y+1}=2 x y $$

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\sqrt{5 x+3 x^{4}} $$

Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a calculator. \(\ln (1.01)\)

Find \(c\) so that \(f^{\prime}(c)=0 .\) . \(f(x)=-x^{2}+1\)

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

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

Differentiate the functions given with respect to the independent variable. $$ h(t)=-\frac{1}{3} t^{4}+4 t $$

In the following examples quantities \(x\) and \(y\) are given. Interpret the role of change dy/dx in words. $$ y \text { is the height of water in a rain collecting column, } x \text { is time. } $$

Find the first and the second derivatives of each function. $$ g(t)=t^{-5 / 2}-t^{1 / 2} $$

In Problems 9-11, find the lines that are (a) tangential and (b) normal to each curve at the given point. $$ x^{2}+y^{2}=25,(4,-3) \text { (circle) } $$

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\frac{1}{\left(x^{3}-1\right)^{4}} $$

Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a calculator. \(e^{0.1}\)

Find \(c\) so that \(f^{\prime}(c)=0 .\) \(f(x)=-x^{2}+4\)

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

Differentiate the functions with respect to the independent variable. \(f(x)=2 x e^{-3 x}\)

Use the product rule to find the derivative with respect to the independent variable. \(f(x)=\left(4-2 x^{2}\right)^{2}\)

Differentiate the functions given with respect to the independent variable. $$ h(t)=\frac{1}{2} t^{2}-3 t+2 $$

Find the first and the second derivatives of each function. $$ f(x)=x^{3}+\frac{1}{x^{3}} $$

In Problems 9-11, find the lines that are (a) tangential and (b) normal to each curve at the given point. $$ \frac{x^{2}}{4}+\frac{y^{2}}{9}=1,\left(1, \frac{3}{2} \sqrt{3}\right) \text { (ellipse) } $$

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\frac{2}{\left(1-2 x^{2}\right)^{3}} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\frac{1}{1+x}\) at \(a=0\)

Find \(c\) so that \(f^{\prime}(c)=0 .\) \(f(x)=(x+2)^{2}\)

Find the derivative with respect to the independent variable. $$ f(x)=\tan (4 x) $$