Question

differentiate y = ln(6 - x - 5x²).

Explanation:

Step1: Recall the chain rule for differentiation.

The chain rule states that if we have a composite function \( y = \ln(u) \) where \( u = 6 - x - 5x^2 \), then the derivative \( y' \) is given by \( \frac{dy}{dx}=\frac{1}{u}\cdot\frac{du}{dx} \).

Step2: Differentiate the outer function.

For \( y = \ln(u) \), the derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \), so \( \frac{dy}{du}=\frac{1}{u}=\frac{1}{6 - x - 5x^2} \).

Step3: Differentiate the inner function.

Now, differentiate \( u = 6 - x - 5x^2 \) with respect to \( x \). The derivative of a constant (6) is 0, the derivative of \( -x \) is \( -1 \), and the derivative of \( -5x^2 \) is \( -10x \) (using the power rule \( \frac{d}{dx}(x^n)=nx^{n - 1} \)). So, \( \frac{du}{dx}=-1 - 10x \).

Step4: Apply the chain rule.

Multiply the derivatives from Step2 and Step3: \( \frac{dy}{dx}=\frac{1}{6 - x - 5x^2}\cdot(-1 - 10x)=\frac{-1 - 10x}{6 - x - 5x^2} \) or we can factor out a negative sign from the numerator to get \( \frac{-(1 + 10x)}{6 - x - 5x^2}=\frac{-10x - 1}{-5x^2 - x + 6} \) (we can also multiply numerator and denominator by -1 to get \( \frac{10x + 1}{5x^2 + x - 6} \), but both forms are correct, and the first form after applying chain rule is also acceptable).

Answer:

\( \frac{-10x - 1}{6 - x - 5x^2} \) (or equivalent simplified forms like \( \frac{10x + 1}{5x^2 + x - 6} \))