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Chapter 3: Problem 30

Verify the following derivative formulas using the Quotient Rule. $$\frac{d}{d x}(\sec x)=\sec x \tan x$$

Short Answer

Expert verified
Question: Verify that the derivative of the secant function, \(\sec x\), is given by the formula \(\frac{d}{dx}(\sec x)=\sec x \tan x\). Answer: We verified the given derivative formula using the quotient rule and found that $$\frac{d}{dx}(\sec x)=\sec x \tan x$$ as claimed.

Step by step solution

01

Rewrite the secant function

We will rewrite the secant function as a quotient. Remember that \(\sec x = \frac{1}{\cos x}\). Now we have a quotient of two functions, with \(f(x) = 1\) and \(h(x) =\cos x\).
02

Apply the quotient rule

Now apply the quotient rule with \(f(x) = 1\) and \(h(x) = \cos x\): $$\frac{d}{dx}(\frac{1}{\cos x})=\frac{\cos x\frac{d(1)}{dx}-1\frac{d(\cos x)}{dx}}{(\cos x)^2}$$
03

Evaluate the derivatives

Find the derivatives of the numerator functions: $$\frac{d(1)}{dx}=0\text{ and }\frac{d(\cos x)}{dx}=-\sin x$$ Then substitute these derivatives back into the equation from step 2: $$\frac{d}{dx}(\frac{1}{\cos x})=\frac{\cos x(0)-1(-\sin x)}{(\cos x)^2}$$
04

Simplify the expression

Simplify the expression to get the final result: $$\frac{d}{dx}(\sec x)=\frac{\sin x}{(\cos x)^2}$$ Now, recall the trigonometric identity \(\tan x = \frac{\sin x}{\cos x}\). Using this identity, we can further rewrite the expression as: $$\frac{d}{dx}(\sec x)=\frac{1}{\cos x} \cdot \frac{\sin x}{\cos x} = \sec x \tan x$$ As our result matches the given formula, we have successfully verified the derivative formula: $$\frac{d}{d x}(\sec x)=\sec x \tan x$$

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Most popular questions from this chapter

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants (see figure). Find \(d y / d x\) for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=m x ; x^{2}+y^{2}=a^{2},\) where \(m\) and \(a\) are constants

a. Identify the inner function \(g\) and the outer function \(f\) for the composition \(f(g(x))=e^{k x},\) where \(k\) is a real number. b. Use the Chain Rule to show that \(\frac{d}{d x}\left(e^{k x}\right)=k e^{k x}\).

Product Rule for three functions Assume that \(f, g,\) and \(h\) are differentiable at \(x\) a. Use the Product Rule (twice) to find a formula for \(\frac{d}{d x}(f(x) g(x) h(x))\) b. Use the formula in (a) to find \(\frac{d}{d x}\left(e^{2 x}(x-1)(x+3)\right)\)

An observer stands \(20 \mathrm{m}\) from the bottom of a 10 -m-tall Ferris wheel on a line that is perpendicular to the face of the Ferris wheel. The wheel revolves at a rate of \(\pi \mathrm{rad} / \mathrm{min},\) and the observer's line of sight with a specific seat on the wheel makes an angle \(\theta\) with the ground (see figure). Forty seconds after that seat leaves the lowest point on the wheel, what is the rate of change of \(\theta ?\) Assume the observer's eyes are level with the bottom of the wheel.

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. $$x^{2}\left(3 y^{2}-2 y^{3}\right)=4$$
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