Math Problem Statement

Find the derivative of the following function, where a and b are nonzero constants: y = (a sin x + b cos x) / (a sin x - b cos x).

Solution

The given function is:

y=asinx+bcosxasinxbcosxy = \frac{a \sin x + b \cos x}{a \sin x - b \cos x}

To find the derivative dydx\frac{dy}{dx}, we can apply the quotient rule for derivatives, which states:

ddx(u(x)v(x))=v(x)u(x)u(x)v(x)[v(x)]2\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{[v(x)]^2}

Here, u(x)=asinx+bcosxu(x) = a \sin x + b \cos x and v(x)=asinxbcosxv(x) = a \sin x - b \cos x.

Step 1: Compute u(x)u'(x) and v(x)v'(x)

  • u(x)=acosxbsinxu'(x) = a \cos x - b \sin x
  • v(x)=acosx+bsinxv'(x) = a \cos x + b \sin x

Step 2: Apply the Quotient Rule

Substituting into the quotient rule formula:

dydx=(asinxbcosx)(acosxbsinx)(asinx+bcosx)(acosx+bsinx)(asinxbcosx)2\frac{dy}{dx} = \frac{(a \sin x - b \cos x)(a \cos x - b \sin x) - (a \sin x + b \cos x)(a \cos x + b \sin x)}{(a \sin x - b \cos x)^2}

Step 3: Simplify the Numerator

Expanding both terms in the numerator:

  1. (asinxbcosx)(acosxbsinx)(a \sin x - b \cos x)(a \cos x - b \sin x)

=a2sinxcosxabsin2xabcos2x+b2cosxsinx= a^2 \sin x \cos x - ab \sin^2 x - ab \cos^2 x + b^2 \cos x \sin x =(a2+b2)sinxcosxab(sin2x+cos2x)= (a^2 + b^2) \sin x \cos x - ab (\sin^2 x + \cos^2 x) Since sin2x+cos2x=1\sin^2 x + \cos^2 x = 1, this simplifies to: =(a2+b2)sinxcosxab= (a^2 + b^2) \sin x \cos x - ab

  1. (asinx+bcosx)(acosx+bsinx)(a \sin x + b \cos x)(a \cos x + b \sin x)

=a2sinxcosx+absin2x+abcos2x+b2cosxsinx= a^2 \sin x \cos x + ab \sin^2 x + ab \cos^2 x + b^2 \cos x \sin x =(a2+b2)sinxcosx+ab= (a^2 + b^2) \sin x \cos x + ab

Step 4: Combine the Two Terms in the Numerator

[ \text{Numerator} = [(a^2 + b^2) \sin x \cos x - ab] - [(a^2 + b^2) \sin x \cos x + ab] ] =2ab= -2ab

Step 5: Final Derivative

Now, we can write the derivative as:

dydx=2ab(asinxbcosx)2\frac{dy}{dx} = \frac{-2ab}{(a \sin x - b \cos x)^2}

This is the required derivative of the given function.

Let me know if you would like more details on any of the steps! Here are some related questions:

  1. How do you apply the quotient rule to a general function?
  2. What is the physical interpretation of the derivative for trigonometric functions?
  3. How do we differentiate products of trigonometric functions?
  4. Can we use the chain rule to derive the result in another way?
  5. What happens if aa or bb becomes zero in this context?

Tip: The quotient rule can simplify complex rational functions but always requires careful application of derivatives for both the numerator and the denominator.

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Math Problem Analysis

Mathematical Concepts

Calculus
Trigonometric Functions
Differentiation

Formulas

Quotient Rule: (d/dx) [u(x)/v(x)] = [v(x)u'(x) - u(x)v'(x)] / [v(x)]^2
Derivative of sin(x): cos(x)
Derivative of cos(x): -sin(x)

Theorems

Quotient Rule
Trigonometric Derivatives

Suitable Grade Level

Grade 12 or College Calculus