Questions: Find the derivative of y=(7x^4-6)^x. Be sure to include parentheses around the arguments of any logarithmic functions in your answer.

Transcript text: Find the derivative of $y=\left(7 x^{4}-6\right)^{x}$. Be sure to include parentheses around the arguments of any logarithmic functions in your answer.

Solution

Solution Steps

To find the derivative of the function $ y = (7x^4 - 6)^x $, we can use logarithmic differentiation. This involves taking the natural logarithm of both sides to simplify the expression, differentiating implicitly, and then solving for $ \frac{dy}{dx} $.

Step 1: Define the Function

We start with the function given by $ y = (7x^4 - 6)^x $.

Step 2: Apply Logarithmic Differentiation

Taking the natural logarithm of both sides, we have: \[ \log(y) = x \log(7x^4 - 6) \]

Step 3: Differentiate Implicitly

Differentiating both sides with respect to $ x $: \[ \frac{1}{y} \frac{dy}{dx} = \log(7x^4 - 6) + x \cdot \frac{28x^3}{7x^4 - 6} \]

Step 4: Solve for $ \frac{dy}{dx} $

Multiplying both sides by $ y $ gives: \[ \frac{dy}{dx} = y \left( \log(7x^4 - 6) + \frac{28x^4}{7x^4 - 6} \right) \] Substituting back $ y = (7x^4 - 6)^x $: \[ \frac{dy}{dx} = (7x^4 - 6)^x \left( \log(7x^4 - 6) + \frac{28x^4}{7x^4 - 6} \right) \]

Step 5: Simplify the Expression

The expression can be simplified to: \[ \frac{dy}{dx} = (7x^4 - 6)^{x - 1} \left( 28x^4 + (7x^4 - 6) \log(7x^4 - 6) \right) \]