91Ó°ÊÓ

In Exercises \(1-56,\) find the derivatives. Assume that \(a\) and \(b\) are constants. $$y=\sqrt{s^{3}+1}$$

Find the limit of the function as \(x \rightarrow \infty\) $$\frac{\sinh \left(x^{2}\right)}{\cosh \left(x^{2}\right)}$$

Find the derivatives of the functions. Assume \(a, b,\) and \(c\) are constants.$$f(x)=2 x \sin (3 x)$$

Suppose \(f\) has a continuous positive second derivative for all \(x\). Which is larger, \(f(1+\Delta x)\) or \(f(1)+f^{\prime}(1) \Delta x ?\) Explain.

Find the derivatives of the functions. Assume that \(a, b, c,\) and \(k\) are constants. $$g(x)=2 x-\frac{1}{\sqrt[3]{x}}+3^{x}-e$$

Find the derivatives of the given functions. Assume that \(a, b, c,\) and \(k\) are constants. $$y=4 x^{3 / 2}-5 x^{1 / 2}$$

Find the slope of the tangent to the curve at the point specified. $$x^{3}+5 x^{2} y+2 y^{2}=4 y+11 \text { at }(1,2)$$

Suppose \(f^{\prime}(x)\) is a differentiable decreasing function fo all \(x\). In each of the following pairs, which number is the larger? Give a reason for your answer. (a) \(f^{\prime}(5)\) and \(f^{\prime}(6)\) (b) \(f^{\prime \prime}(5)\) and 0 (c) \(f(5+\Delta x)\) and \(f(5)+f^{\prime}(5) \Delta x\)

Prove that if \(f^{\prime}(x)=g^{\prime}(x)\) for all \(x\) in \((a, b),\) then there is a constant \(C\) such that \(f(x)=g(x)+C\) on \((a, b) .\) [Hint: Apply the Constant Function Theorem to \(h(x)=f(x)-g(x) .]\)

Find the derivatives of the functions. Assume \(a, b,\) and \(c\) are constants.$$y=e^{\theta} \sin (2 \theta)$$