91影视

Find the derivative of \(y=4 t^{12}-7 \sqrt{t}+\frac{4}{t}\). \(\frac{d y}{d t}=\) _________

Let \(f(x)=a^{x}\). The goal of this problem is to explore how the value of \(a\) affects the derivative of \(f(x)\), without assuming we know the rule for \(\frac{d}{d x}\left[a^{x}\right]\) that we have stated and used in earlier work in this section. a. Use the limit definition of the derivative to show that $$ f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{a^{x} \cdot a^{h}-a^{x}}{h}. $$ b. Explain why it is also true that $$ f^{\prime}(x)=a^{x} \cdot \lim _{h \rightarrow 0} \frac{a^{h}-1}{h}. $$ c. Use computing technology and small values of \(h\) to estimate the value of $$ L=\lim _{h \rightarrow 0} \frac{a^{h}-1}{h} $$ when \(a=2\). Do likewise when \(a=3\). d. Note that it would be ideal if the value of the limit \(L\) was \(1,\) for then \(f\) would be a particularly special function: its derivative would be simply \(a^{x},\) which would mean that its derivative is itself. By experimenting with different values of \(a\) between 2 and 3, try to find a value for \(a\) for which $$ L=\lim _{h \rightarrow 0} \frac{a^{h}-1}{h}=1. $$ e. Compute \(\ln (2)\) and \(\ln (3)\). What does your work in (b) and (c) suggest is true about \(\frac{d}{d x}\left[2^{x}\right]\) and \(\frac{d}{d x}\left[3^{x}\right] ?\) f. How do your investigations in (d) lead to a particularly important fact about the function \(f(x)=e^{x} ?\)

Find the equation of the tangent line to the curve \(y=2 \tan x\) at the point \((\pi / 4,2)\). The equation of this tangent line can be written in the form \(y=m x+b\) where \(m\) is:__________ and where \(b\) is: ____________.

For the curve given by the equation \(\sin (x+y)+\cos (x-y)=1,\) find the equation of the tangent line to the curve at the point \(\left(\frac{\pi}{2}, \frac{\pi}{2}\right)\).

Use implicit differentiation to find an equation of the tangent line to the curve \(x y^{3}+4 x y=\) 40 at the point (8,1). The equation \(\square\) defines the tangent line to the curve at the point (8,1) .