Question

Evaluate each expression. $$\frac{d}{d x}\left(3 x^{5}+2 x\right)$$

Short answer

The derivative of the expression \(3x^{5} + 2x\) with respect to \(x\) is \(15x^4 + 2\).

Step-by-step solution

Apply the Power Rule

To find the derivative of the given polynomial, apply the power rule to each term separately. The power rule states that if you have a term in the form of \(ax^n\), its derivative is \(n * ax^{n-1}\). Use this rule on the terms \(3x^5\) and \(2x^1\).

Differentiate each term

Differentiate the first term \(3x^5\) by bringing down the exponent and reducing it by one to get \(5 * 3x^{5-1} = 15x^4\). Similarly, the derivative of \(2x\) is \(1 * 2x^{1-1} = 2x^0\), since any nonzero number raised to the power of 0 is 1.

Simplify the Result

Combine the results of the derivatives of both terms to obtain the overall derivative. Since \(x^0 = 1\), \(2x^0\) simplifies to 2. The final derivative is then \(15x^4 + 2\).

Key concepts

Power Rule Differentiation

Understanding power rule differentiation is essential for anyone studying calculus. It's a simple yet powerful tool that helps in finding derivatives quickly and efficiently. The power rule applies to terms in the form of ( ax^n ), where a is a constant and n is a real number exponent.

When applying the power rule, you multiply the term by the exponent and then subtract one from the exponent. Mathematically, it's represented as \( \frac{d}{dx}(ax^n) = n \cdot ax^{n-1} \). For example, to differentiate 3x^5, you would calculate 5 \cdot 3x^{5-1} = 15x^4.

This rule makes it significantly easier to tackle polynomials, as you can differentiate term by term without needing complex calculations. Remember, this rule can be applied when n is any real number, which gives it a wide range of application in differential calculus.

Polynomial Differentiation

Polynomial differentiation involves taking the derivative of polynomial functions which are sums of terms like ax^n. A polynomial can have many terms, but the beauty of differentiation is that you can handle each term separately.

In the exercise provided, the polynomial 3x^5 + 2x is differentiated term by term. For the first term 3x^5, we use power rule differentiation to get 15x^4. For the second term 2x, which can be written as 2x^1, the power rule yields 2x^0, which simplifies further to 2, since any term raised to the power of zero equals one.

When differentiating polynomials, it's important to work methodically, differentiating and simplifying each term before combining them back together for the final derivative.

Simplifying Derivatives

The final step of differentiation often involves simplifying the expression to its most basic form. Simplifying derivatives allows for a clearer understanding and easier application of the calculated derivative. After applying the power rule and finding the derivatives of each term separately, as done in the exercise, we combine them to form the overall derivative.

In our example, after applying the power rule, we get 15x^4 and 2x^0. We know that x^0 simplifies to 1, thus 2x^0 simply becomes 2. Combining these results gives us the simplified form of the derivative as 15x^4 + 2.

This step not only involves arithmetic simplification but also a keen understanding of algebraic principles. Remember that the final derivative should be as simple as possible, which often involves consolidating like terms and eliminating any mathematical redundancy.