91Ó°ÊÓ

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. \(\sin \left(\frac{\pi}{2}+0.02\right)\)

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.

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

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

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

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

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