91Ó°ÊÓ

Derivatives from limits The following limits represent \(f^{\prime}(a)\) for some function \(f\) and some real number \(a\) a. Find a possible function \(f\) and number \(a\). b. Evaluate the limit by computing \(f^{\prime}(a)\). $$\lim _{x \rightarrow 1} \frac{x^{100}-1}{x-1}$$

Logarithmic differentiation Use logarithmic differentiation to evaluate \(f^{\prime}(x)\). $$f(x)=\left(1+x^{2}\right)^{\sin x}$$

Find the derivative of the inverse cosine function in the following two ways. a. Using Theorem 3.21 b. Using the identity \(\sin ^{-1} x+\cos ^{-1} x=\pi / 2\)

Assume both the graphs of f and g pass through the point \((3,2), f^{\prime}(3)=5,\) and \(g^{\prime}(3)=-10 .\) If \(p(x)=f(x) g(x)\) and \(q(x)=f(x) / g(x),\) find the following derivatives. $$q^{\prime}(3)$$

Logarithmic differentiation Use logarithmic differentiation to evaluate \(f^{\prime}(x)\). $$f(x)=\left(1+\frac{1}{x}\right)^{x}$$

Use a trigonometric identity to show that the derivatives of the inverse cotangent and inverse cosecant differ from the derivatives of the inverse tangent and inverse secant, respectively, by a multiplicative factor of -1.

Derivatives from limits The following limits represent \(f^{\prime}(a)\) for some function \(f\) and some real number \(a\) a. Find a possible function \(f\) and number \(a\). b. Evaluate the limit by computing \(f^{\prime}(a)\). $$\lim _{h \rightarrow 0} \frac{\sqrt{9+h}-\sqrt{9}}{h}$$

Let$$g(x)=\left\\{\begin{array}{ll}\frac{1-\cos x}{2 x} & \text { if } x \neq 0 \\\a & \text { if } x=0\end{array}\right.$$ For what values of \(a\) is \(g\) continuous?

Let \(f(x)=x e^{2 x}\) a. Find the values of \(x\) for which the slope of the curve \(y=f(x)\) is 0 b. Explain the meaning of your answer to part (a) in terms of the graph of \(f\)

Given that \(f(1)=2\) and \(f^{\prime}(1)=2,\) find the slope of the curve \(y=x f(x)\) at the point \((1,2).\)