Chapter 4

Use the product rule to find the derivative with respect to the independent variable. $$ f(x)=3\left(x^{2}+2\right)\left(4 x^{2}-5 x^{4}\right)-3 $$

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

Find } c \text { so that } f^{\prime}(c)=0 . $$ $$ f(x)=-3 x^{2}+1 $$

Differentiate the functions with respect to the independent variable. \(f(x)=\frac{1}{\left(x^{3}-2\right)^{4}}\)

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

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

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

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

Use (4.12) to find the derivative of the inverse at the indicated point. Let $$f(x)=\sqrt{x+1}, \quad x \geq 0$$ Find \(\left.\frac{d}{d x} f^{-1}(x)\right|_{x=2} .[\) Note that \(f(3)=2 .\) ]

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

Differentiate the functions with respect to the independent variable. \(f(x)=\frac{2}{\left(1-5 x^{2}\right)^{3}}\)

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

Differentiate the functions in Problems 1-52 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 in Problems with respect to the independent variable.$$ h(t)=\frac{1}{2} t^{2}-3 t+2 $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=\frac{1}{1+x} \text { at } a=0 $$

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

Differentiate the functions with respect to the independent variable. \(f(x)=\frac{3 x-1}{\sqrt{2 x^{2}-1}}\)

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

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

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

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=x^{2} \sin \frac{\pi}{3}+\tan \frac{\pi}{4} $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=\frac{1}{1-x} \text { at } a=0 $$

Find } c \text { so that } f^{\prime}(c)=0 . $$ $$ f(x)=(x+3)^{2} $$

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

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

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

Use the product rule to find the derivative with respect to the independent variable. $$ f(x)=\frac{\left(2 x^{2}-3 x+1\right)^{2}}{4}+2 $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=2 x^{3} \cos \frac{\pi}{3}+\cos \frac{\pi}{6} $$

Use (4.12) to find the derivative of the inverse at the indicated point. Let $$f(x)=x+\ln (x+1), \quad x>-1$$ Find \(\left.\frac{d}{d x} f^{-1}(x)\right|_{x=0} .[\) Note that \(f(0)=0 .]\)

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=\frac{2}{1+x} \text { at } a=1 $$

Find } c \text { so that } f^{\prime}(c)=0 . $$ $$ f(x)=x^{2}-6 x+9 $$

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

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

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

Use the product rule to find the derivative with respect to the independent variable. $$ g(s)=\left(2 s^{2}-5 s\right)^{2} $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=-3 x^{4} \tan \frac{\pi}{6}-\cot \frac{\pi}{6} $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=\frac{1}{3-2 x} \text { at } a=2 $$

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

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

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=-3 \csc (3-5 x) $$

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

Use the product rule to find the derivative with respect to the independent variable. $$ h(t)=4\left(3 t^{2}-1\right)(2 t+1) $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=x^{2} \sec \frac{\pi}{6}+3 x \sec \frac{\pi}{4} $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=\frac{1}{(1+x)^{2}} \text { at } a=0 $$

Find } c \text { so that } f^{\prime}(c)=0 . $$ $$ f(x)=\sin \left(\frac{\pi}{2} x\right) $$

Differentiate the functions with respect to the independent variable. \(f(s)=\sqrt{s+\sqrt{s}}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=3 \sin \left(x^{2}\right) $$

Use the product rule to find the derivative with respect to the independent variable. $$ g(t)=3\left(2 t^{2}-5 t^{4}\right)^{2} $$