91Ó°ÊÓ

Let \(f(x)=x^{5}\) and \(g(x)=2 x-3\) (a) Find \((f \circ g)(x)\) and \((f \circ g)^{\prime}(x)\) (b) Find \((g \circ f)(x)\) and \((g \circ f)^{\prime}(x)\)

Compute the derivative of the given function \(f(x)\) by \((a)\) multiplying and then differentiating and (b) using the product rule. Verify that (a) and (b) yield the same result. \(f(x)=\left(x^{2}+1\right)\left(x^{2}-1\right)\)

(a) If you are given an equation for the tangent line at the point \((a, f(a))\) on a curve \(y=f(x),\) how would you go about finding \(f^{\prime}(a) ?\) (b) Given that the tangent line to the graph of \(y=f(x)\) at the point \((2,5)\) has the equation \(y=3 x-1,\) find \(f^{\prime}(2) .\) (c) For the function \(y=f(x)\) in part (b), what is the instantaneous rate of change of \(y\) with respect to \(x\) at \(x=2 ?\)

Let \(f(x)=5 \sqrt{x}\) and \(g(x)=4+\cos x\) (a) Find \((f \circ g)(x)\) and \((f \circ g)^{\prime}(x)\) (b) Find \((g \circ f)(x)\) and \((g \circ f)^{\prime}(x)\)

Given that the tangent line to \(y=f(x)\) at the point \((1,2)\) passes through the point \((-1,-1),\) find \(f^{\prime}(1) .\)

Find \(f^{\prime}(x)\) $$ f(x)=2 \sin ^{2} x $$

Find \(d y / d x\) $$ y=\frac{1}{2}\left(x^{4}+7\right) $$

Compute the derivative of the given function \(f(x)\) by \((a)\) multiplying and then differentiating and (b) using the product rule. Verify that (a) and (b) yield the same result. \(f(x)=(x+1)\left(x^{2}-x+1\right)\)

Find \(f^{\prime}(x)\). \(f(x)=\left(3 x^{2}+6\right)\left(2 x-\frac{1}{4}\right)\)

If a particle moves at constant velocity, what can you say about its position versus time curve?