Question

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=\sqrt[3]{6 x^{2}+1} $$

Short answer

The derivative of the function \(y = \sqrt[3]{6x^{2}+1}\) is \( y' = 4x(6x^2+1)^{-2/3}\).

Step-by-step solution

Converting to Exponential Form

Rewrite the function \(y=\sqrt[3]{6 x^{2}+1}\) into exponential form to enable application of power rule of differentiation. The cube root can be written as power of \(1/3\). So, \(y=(6 x^{2}+1)^{(1/3)}\).

Applying Power Rule and Chain Rule

Apply the power rule of differentiation first, by bringing the power \(1/3\) down as a coefficient, lowering the power by one to get \(-2/3\), and then applying the chain rule to the inner function. So, \( y' = (1/3)(6 x^{2}+1)^{-2/3} \cdot (12x)\).

Simplify the Derivative

Simplifying the derivative expression, results in \( y' = 4x(6x^2+1)^{-2/3}\).

Key concepts

Power Rule Differentiation

Understanding the power rule is essential for students tackling calculus problems, as it's a fundamental technique used to find derivatives. This rule simplifies the differentiation process when you're dealing with powers of x. The basic idea is that if you have a function in the form of f(x) = x^n, where n is any real number, the derivative f'(x) is given by nx^(n-1).

The power rule can be applied to any term in the function individually, regardless of whether n is positive, negative, or a fraction. Take the function from the exercise, y = (6x^2 + 1)^(1/3). The term (6x^2 + 1) is raised to the power of 1/3, so applying the power rule directly would involve multiplying by 1/3 and then subtracting 1 from the exponent, yielding (1/3)(6x^2 + 1)^(-2/3). This is a clear example of how power rule simplifies the process of finding derivatives.

Chain Rule Differentiation

When you're dealing with composite functions, the chain rule comes into play. It's a bit like peeling an onion, where each layer corresponds to a function within another function. The rule allows us to differentiate the outer function and the inner function in a specific sequence.

In the context of our exercise, the overall function is (6x^2 + 1)^(1/3), where (6x^2 + 1) is the inner function and the outer function is the cube root operation. According to the chain rule, we differentiate the outer function first, as we did with the power rule, and then multiply it by the derivative of the inner function, which is 12x. Combining these, we get (1/3)(6x^2 + 1)^(-2/3) · (12x), demonstrating how the chain rule allows us to find the derivative of complex, layered functions with ease.

Simplifying Derivatives

Once we apply rules like power rule and chain rule, we often end up with expressions that can appear complicated and difficult to understand. Simplifying derivatives plays a critical role in making these expressions clearer and easier to work with. It involves combining like terms, factoring, and sometimes even expanding expressions to their simplest form.

In our exercise, after applying the power rule and the chain rule, we obtained (1/3)(6x^2 + 1)^(-2/3) · (12x). To simplify this expression, we performed multiplication and consolidated the constants, resulting in 4x(6x^2 + 1)^(-2/3). This final form is much clearer and compact, showing that simplifying can greatly aid in understanding the results of differentiation and preparing for further calculus operations.