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Chapter 2: Problem 13

Find \(d y / d x\) by implicit differentiation. $$ \sin x+2 \cos 2 y=1 $$

Short Answer

Expert verified
\(dy/dx = \cos x / (4 \sin 2y)\)

Step by step solution

01

Differentiate Both Sides of the Equation

Differentiate both sides of the given equation with respect to \(x\). The derivative of \(\sin x\) with respect to \(x\) is \(\cos x\). The derivative of \(2 \cos 2y\) is \(2(-2 \sin 2y)(dy/dx)\), where \(dy/dx\) comes from the chain rule when applying the derivative to the \(y\) term. The derivative of the constant 1 is zero. So, the differentiated equation is: \(\cos x - 4 \sin 2y (dy/dx) = 0\).
02

Solve for \(dy/dx\)

Now isolate \(dy/dx\). Move the \(\cos x\) term to the other side of the equation, then divide all terms by the coefficient of \(dy/dx\) to isolate it, obtaining: \(dy/dx = \cos x / (4 \sin 2y)\).
03

Simplify the Expression

Simplifying the right-hand side results in the final expression for the derivative: \(dy/dx = \cos x / (4 \sin 2y)\).

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Most popular questions from this chapter

In Exercises 15-28, find the derivative of the function. $$ y=\arctan \frac{x}{2}-\frac{1}{2\left(x^{2}+4\right)} $$

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{y=\cos 3 x} \quad \frac{\text { Point }}{\left(\frac{\pi}{4},-\frac{\sqrt{2}}{2}\right)}\)

In Exercises 37 and 38 , the derivative of the function has the same sign for all \(x\) in its domain, but the function is not one-to-one. Explain. $$ f(x)=\frac{x}{x^{2}-4} $$

Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. \(g(t)=\tan 2 t, \quad\left(\frac{\pi}{6}, \sqrt{3}\right)\)

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{y=4-x^{2}-\ln \left(\frac{1}{2} x+1\right)} \quad \frac{\text { Point }}{\left(0,4\right)}\)
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