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Chapter 3: Problem 40

Find the derivative of the following functions. $$y=\csc ^{2} \theta-1$$

Short Answer

Expert verified
Question: Find the derivative of the function \(y = \csc^2\theta - 1\) with respect to \(\theta\). Answer: \(y' = -2\csc^2\theta\cot\theta\)

Step by step solution

01

Differentiate the first term#csc^2\theta

Recall that the derivative of \(\csc \theta\) with respect to \(\theta\) is $$-\csc \theta \cot \theta$$ Now, applying the chain rule, differentiate \(\csc^2\theta\) with respect to \(\theta\): $$\frac{d}{d\theta}(\csc^2\theta) = 2(\csc\theta)(-\csc\theta\cot\theta) = -2\csc^2\theta\cot\theta$$
02

Differentiate the second term#-1

The derivative of a constant term is always 0. Therefore, the derivative of the constant term \(-1\) is: $$\frac{d}{d\theta}(-1) = 0$$
03

Combine the derivatives of both terms

Finally, sum the derivatives of the first and second terms to get the derivative of the entire function: $$y' = \frac{d}{d\theta}(y) = \frac{d}{d\theta}(\csc^2\theta - 1) = -2\csc^2\theta\cot\theta + 0$$ The derivative of the given function is: $$y' = -2\csc^2\theta\cot\theta$$

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Most popular questions from this chapter

Suppose \(f\) is differentiable on an interval containing \(a\) and \(b\), and let \(P(a, f(a))\) and \(Q(b, f(b))\) be distinct points on the graph of \(f\). Let \(c\) be the \(x\) -coordinate of the point at which the lines tangent to the curve at \(P\) and \(Q\) intersect, assuming that the tangent lines are not parallel (see figure). a. If \(f(x)=x^{2},\) show that \(c=(a+b) / 2,\) the arithmetic mean of \(a\) and \(b\), for real numbers \(a\) and \(b\) b. If \(f(x)=\sqrt{x}\), show that \(c=\sqrt{a b}\), the geometric mean of \(a\) and \(b\), for \(a > 0\) and \(b > 0\) c. If \(f(x)=1 / x,\) show that \(c=2 a b /(a+b),\) the harmonic mean of \(a\) and \(b,\) for \(a > 0\) and \(b > 0\) d. Find an expression for \(c\) in terms of \(a\) and \(b\) for any (differentiable) function \(f\) whenever \(c\) exists.

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

The position (in meters) of a marble rolling up a long incline is given by \(s=\frac{100 t}{t+1},\) where \(t\) is measured in seconds and \(s=0\) is the starting point. a. Graph the position function. b. Find the velocity function for the marble. c. Graph the velocity function and give a description of the motion of the marble. d. At what time is the marble 80 m from its starting point? e. At what time is the velocity \(50 \mathrm{m} / \mathrm{s} ?\)

Calculating limits exactly Use the definition of the derivative to evaluate the following limits. $$\lim _{x \rightarrow e} \frac{\ln x-1}{x-e}$$

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{d x}(\ln x)^{x^{2}}$$
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