91Ó°ÊÓ

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

In Problems \(21-24\), apply the product rule to find the normal line, in slope- intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=(1-x)\left(2-x^{2}\right), \text { at } x=2 $$

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

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

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=-5 \cos \left(2-x^{3}\right)+2 \cos ^{3}(x-4) $$

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

Differentiate the functions with respect to the independent variable. \(h(x)=\sqrt[3]{1-2 x}\)

Use the formal definition to find the derivative of \(y=\) \(-2 x^{2}\) at \(x=1\) (b) Show that the point \((1,-2)\) is on the graph of \(y=-2 x^{2}\), and find the equation of the tangent line at the point \((1,-2)\). (c) Graph \(y=-2 x^{2}\) and the tangent line at the point \((1,-2)\) in the same coordinate system.

Apply the product rule to find the normal line, in slope-intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=5(1-2 x)(x+1)-3, \text { at } x=0 $$

Differentiate the functions with respect to the independent variable. \(f(x)=\sqrt[7]{x^{2}-2 x+1}\)