91Ó°ÊÓ

Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. $$y=5\left(7 x^{3}+1\right)^{-3}$$

Evaluate the derivatives of the following functions. $$f(y)=\cot ^{-1}\left(1 /\left(y^{2}+1\right)\right)$$

Consider the following cost functions. a. Find the average cost and marginal cost functions. b. Determine the average and marginal cost when \(x=a\). c. Interpret the values obtained in part (b). $$C(x)=-0.01 x^{2}+40 x+100,0 \leq x \leq 1500, a=1000$$

Find the derivative of the following functions. $$s(t)=4 \sqrt{t}-\frac{1}{4} t^{4}+t+1$$

Derivatives of \(b^{x}\) Find the derivatives of the following functions. $$y=5^{3 t}$$

Equations of tangent lines by definition (2) a. Use definition (2) ( \(p .\) 129) to find the slope of the line tangent to the graph of \(f\) at \(P\). b. Determine an equation of the tangent line at \(P\). $$f(x)=\sqrt{x-1} ; P(2,1)$$

Use implicit differentiation to find \(\frac{d y}{d x}\) $$\sqrt{x+y^{2}}=\sin y$$

A line perpendicular to another line or to a tangent line is called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point \(P\). $$y=x^{2}-3 x ; P(3,0)$$

Consider the following cost functions. a. Find the average cost and marginal cost functions. b. Determine the average and marginal cost when \(x=a\). c. Interpret the values obtained in part (b). $$C(x)=-0.04 x^{2}+100 x+800,0 \leq x \leq 1000, a=500$$

Use Version 2 of the Chain Rule to calculate the derivatives of the following functions. $$y=\cos 5 t$$