Problem 30 - Single Variable Calculus

Question

Find the derivative of the function. \(f(t)=2 t^{2}+\sqrt{t^{3}}\)

Step-by-Step Solution

Answer

The derivative of the function, \(f(t) = 2t^2 + \sqrt{t^3}\), is: \(f'(t) = 4t + \frac{3}{2}t^{2 + 3(1/2)}\).

Step 1: Identify terms to apply power rule and chain rule

Our function is \(f(t) = 2t^2 + \sqrt{t^3}\), which can be rewritten as \(f(t) = 2t^2 + (t^3)^{1/2}\). We can apply the power rule directly to the first term and the chain rule to the second term.

Step 2: Apply the power rule to the first term

For the first term, \(2t^2\), we have an exponent of 2. So applying the power rule, we get: \[\frac{d}{dt}(2t^2) = 2(2t^{2-1}) = 4t\]

Step 3: Apply the chain rule to the second term

For the second term, \((t^3)^{1/2}\), we have a composite function of the form \(f(g(t))\), where \(f(u) = u^{1/2}\) and \(g(t) = t^3\). We will first find the derivatives of the inner and outer functions individually. The derivative of the outer function with respect to its argument, u, is: \[\frac{d}{du}(u^{1/2}) = \frac{1}{2}u^{-1/2}\] The derivative of the inner function with respect to t is: \[\frac{d}{dt}(t^3) = 3t^2\] Now, applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function: \[\frac{d}{dt}((t^3)^{1/2}) = \frac{1}{2}(t^3)^{-1/2}(3t^2)\]

Step 4: Simplify the second term's derivative

Simplifying the expression for the second term's derivative: \[\frac{1}{2}(t^3)^{-1/2}(3t^2) = \frac{3}{2}t^{3(1/2)}t^2\]

Step 5: Combine the derivatives and simplify

Now, we add the derivatives of the first and second terms to get the derivative of the entire function: \(f'(t) = 4t + \frac{3}{2}t^{3(1/2)}t^2\) Simplifying the expression: \[f'(t) = 4t + \frac{3}{2}t^{2 + 3(1/2)}\] So, the derivative of the function, \(f(t) = 2t^2 + \sqrt{t^3}\), is: \[f'(t) = 4t + \frac{3}{2}t^{2 + 3(1/2)}\]

Key Concepts

Power Rule

The Power Rule is one of the most straightforward and commonly used differentiation rules. It helps us find the derivative of a term that includes a variable raised to a power. In general, for a function of the form \(x^n\), the derivative is \(nx^{n-1}\).

In our exercise, the first term of the function is \(2t^2\). To apply the power rule, we multiply the coefficient by the power of the variable, resulting in \(2 \times 2\), and decrease the exponent by one. Therefore, the derivative of \(2t^2\) becomes \(4t\). This process highlights the power rule's efficiency in simplifying polynomial terms.

• Identify the base and exponent.
• Multiply the exponent by the coefficient.
• Subtract one from the exponent.

Chain Rule

The Chain Rule is a powerful tool when dealing with composite functions. A composite function is made from two or more functions, symbolically written as \(f(g(x))\). The chain rule helps us differentiate such expressions by linking the derivative of the outer function with that of the inner function.

For \(\sqrt{t^3}\), it helps to rewrite it as \((t^3)^{1/2}\), making it easier to visualize it as a composition. The outer function \(f(u) = u^{1/2}\) and the inner function \(g(t) = t^3\) can then be differentiated separately: the derivative of \(f\) with respect to \(u\) is \(\frac{1}{2}u^{-1/2}\), and the derivative of \(g\) with respect to \(t\) is \(3t^2\).

• Differentiating the outer function.
• Differentiating the inner function.
• Multiplying these two derivatives together.

Composite Function

Composite functions involve an inner and outer function combined such that the output of one function becomes the input of another. This is key in situations where functions are nested within each other, requiring the use of the chain rule.

In our given function, the term \((t^3)^{1/2}\) is a composite of two functions. Recognizing this allows us to apply the chain rule effectively, which is integral to solving the derivative. By finding the derivative of the outer function relative to the inner one, and multiplying it with the derivative of the inner function, we arrive at the desired result.

• Recognize the inner and outer functions clearly.
• Utilize the chain rule to separate the components.
• Ensure all derivatives are calculated before recombining.

Simplification of Terms

Simplification is a crucial step in calculus to make expressions more manageable and easier to interpret. After applying differentiation rules, it often results in complex expressions that require combination and reduction.

When dealing with the derivative, the final step involves simplifying all collected terms. From our exercise, upon getting \(f'(t) = 4t + \frac{3}{2}t^{2 + 3(1/2)}\), simplifying powers of \(t\) terms helps standardize the expression. Combining terms, where possible, provides clarity and finality to solutions, ensuring they are tidy and interpretable.

• Organize like terms together for consolidation.
• Simplify any fractions, if feasible.
• Recheck each step for coherence and accuracy.