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Chapter 3: Problem 32

Find the derivative of the following functions. $$y=\tan x+\cot x$$

Short Answer

Expert verified
Answer: The derivative of the function $$y = \tan x + \cot x$$ is $$\frac{d}{dx}(y) = \sec^2 x - \csc^2 x$$.

Step by step solution

01

Derivative of the tangent function

To find the derivative of the tangent function, we will use the definition: \(\frac{d}{dx}(\tan x) = \sec^2 x\). Therefore, the derivative of the tangent function is: $$ \frac{d}{dx}(\tan x)=\sec^2 x $$
02

Derivative of the cotangent function

To find the derivative of the cotangent function, we will use the definition: \(\frac{d}{dx}(\cot x) = -\csc^2 x\). Therefore, the derivative of the cotangent function is: $$ \frac{d}{dx}(\cot x)=-\csc^2 x $$
03

Apply the sum rule

Now, we can find the derivative of the entire function by applying the sum rule, according to which: \(\frac{d}{dx}(f(x)+g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))\). In our case, \(f(x)=\tan x\) and \(g(x)=\cot x\). The derivative of \(y(\tan x+\cot x)\) is equal to the sum of the derivatives of the functions found in steps 1 and 2: $$ \frac{d}{dx}(y)=\frac{d}{dx}(\tan x+\cot x)=\frac{d}{dx}(\tan x) + \frac{d}{dx}(\cot x)=\sec^2 x -\csc^2 x $$
04

Write the final answer

The derivative of the given function $$y = \tan x + \cot x$$ is: $$ \frac{d}{dx}(y) = \sec^2 x - \csc^2 x $$

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Most popular questions from this chapter

Vertical tangent lines a. Determine the points at which the curve \(x+y^{3}-y=1\) has a vertical tangent line (see Exercise 52 ). b. Does the curve have any horizontal tangent lines? Explain.

Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. $$x\left(1-y^{2}\right)+y^{3}=0$$

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{d x}\left(x^{\pi}+\pi^{x}\right)$$

A 500-liter (L) tank is filled with pure water. At time \(t=0,\) a salt solution begins flowing into the tank at a rate of \(5 \mathrm{L} / \mathrm{min} .\) At the same time, the (fully mixed) solution flows out of the tank at a rate of \(5.5 \mathrm{L} / \mathrm{min}\). The mass of salt in grams in the tank at any time \(t \geq 0\) is given by $$M(t)=250(1000-t)\left(1-10^{-30}(1000-t)^{10}\right)$$ and the volume of solution in the tank (in liters) is given by \(V(t)=500-0.5 t\). a. Graph the mass function and verify that \(M(0)=0\). b. Graph the volume function and verify that the tank is empty when \(t=1000\) min. c. The concentration of the salt solution in the tank (in \(\mathrm{g} / \mathrm{L}\) ) is given by \(C(t)=M(t) / V(t) .\) Graph the concentration function and comment on its properties. Specifically, what are \(C(0)\) and \(\lim _{\theta \rightarrow 000^{-}} C(t) ?\) \(t \rightarrow 1\) d. Find the rate of change of the mass \(M^{\prime}(t),\) for \(0 \leq t \leq 1000\). e. Find the rate of change of the concentration \(C^{\prime}(t),\) for \(0 \leq t \leq 1000\). f. For what times is the concentration of the solution increasing? Decreasing?

Use the properties of logarithms to simplify the following functions before computing \(f^{\prime}(x)\). $$f(x)=\log _{2} \frac{8}{\sqrt{x+1}}$$
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