If `y=e^(-x)``[-6a sin ax +5b cos bx-6 cos bx) `then `(dy)/(dx)=`

Options

A. `(dy)/(dx) =e^(-x)[-6a sin ax+5b cos bx -6a cos ax-5 sin bx]`
B. `(dy)/(dx) =e^(-x)[6a sin ax-5b cos bx -6a cos ax-5 sin bx]`
C. `(dy)/(dx) =e^(-x)[-6a sin ax+5b cos bx +6 cos ax+5 sin bx]`
D. None of these

Answer

To find the derivative \( \frac{dy}{dx} \) of the function

\[ y = e^{-x} \left(-6a \sin(ax) + 5b \cos(bx) - 6 \cos(bx)\right), \]

we will use the product rule of differentiation. The product rule states that if you have two functions \( u \) and \( v \), then the derivative is given by:

\[ \frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}. \]

### Step 1: Identify \( u \) and \( v \)

Let:
- \( u = e^{-x} \)
- \( v = -6a \sin(ax) + 5b \cos(bx) - 6 \cos(bx) \)

### Step 2: Differentiate \( u \) and \( v \)

1. Differentiate \( u \):
\[ \frac{du}{dx} = \frac{d}{dx}(e^{-x}) = -e^{-x} \]

2. Differentiate \( v \):
\[ \frac{dv}{dx} = \frac{d}{dx}(-6a \sin(ax)) + \frac{d}{dx}(5b \cos(bx)) - \frac{d}{dx}(6 \cos(bx)) \]

- For \( -6a \sin(ax) \):
\[ \frac{d}{dx}(-6a \sin(ax)) = -6a \cdot a \cos(ax) = -6a^2 \cos(ax) \]
- For \( 5b \cos(bx) \):
\[ \frac{d}{dx}(5b \cos(bx)) = 5b \cdot (-b \sin(bx)) = -5b^2 \sin(bx) \]
- For \( -6 \cos(bx) \):
\[ \frac{d}{dx}(-6 \cos(bx)) = -6 \cdot (-b \sin(bx)) = 6b \sin(bx) \]

Combining these, we get:
\[ \frac{dv}{dx} = -6a^2 \cos(ax) - 5b^2 \sin(bx) + 6b \sin(bx) \]
Simplifying:
\[ \frac{dv}{dx} = -6a^2 \cos(ax) + (6b - 5b^2) \sin(bx) \]

### Step 3: Apply the Product Rule

Now we can apply the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

Substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):
\[ \frac{dy}{dx} = e^{-x} \left(-6a^2 \cos(ax) + (6b - 5b^2) \sin(bx)\right) + \left(-6a \sin(ax) + 5b \cos(bx) - 6 \cos(bx)\right)(-e^{-x}) \]

### Step 4: Simplify the Expression

This simplifies to:
\[ \frac{dy}{dx} = e^{-x} \left(-6a^2 \cos(ax) + (6b - 5b^2) \sin(bx) + 6a \sin(ax) - 5b \cos(bx) + 6 \cos(bx)\right) \]

### Final Result

Thus, the final expression for \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = e^{-x} \left(-6a^2 \cos(ax) + 6a \sin(ax) + (6b - 5b^2) \sin(bx) + (6 - 5b) \cos(bx)\right) \]