D/dx(4x^3 - 2x + 6) = 12x^2 - 2

Definition

This expression represents the derivative of the polynomial function $$4x^3 - 2x + 6$$ with respect to $$x$$, resulting in $$12x^2 - 2$$. Derivatives measure the rate at which a function changes and are foundational in understanding how functions behave. This process involves applying rules such as the power rule and constant rule, which simplify the differentiation of polynomial terms.

5 Must Know Facts For Your Next Test

1. In the expression, the derivative is taken term by term, using both the power rule for polynomial terms and recognizing that the derivative of a constant is zero.
2. For the term $$4x^3$$, applying the power rule gives $$12x^2$$ because you multiply the coefficient (4) by the exponent (3) and then reduce the exponent by one.
3. The term $$-2x$$ differentiates to $$-2$$, which comes from applying the power rule where the coefficient stays the same and the exponent (1) decreases by one.
4. The constant term $$6$$ has a derivative of zero, as constants do not change regardless of the value of $$x$$.
5. Combining all these results leads to the complete derivative, which is simplified to $$12x^2 - 2$$.

Review Questions

• How does applying the power rule help in finding the derivative of each term in the expression $$4x^3 - 2x + 6$$?
  • Applying the power rule allows us to easily find derivatives for polynomial terms. For instance, with the term $$4x^3$$, we use the power rule to multiply by the exponent (3) and decrease it by one, resulting in $$12x^2$$. Similarly, for the linear term $$-2x$$, we recognize it as $$-2x^1$$ and apply the rule to find its derivative as $$-2$$. This method simplifies our calculations for each term effectively.
• Why do constants like $$6$$ not contribute to the derivative when differentiating polynomial functions?
  • Constants do not contribute to the derivative because their value does not change regardless of changes in other variables like $$x$$. The derivative measures how a function responds to changes in its input, so when differentiating a constant like $$6$$, we find that it results in zero. This is a key aspect when differentiating polynomials, as only terms with variables will have non-zero derivatives.
• Evaluate how understanding this derivative impacts your ability to analyze more complex functions or real-world scenarios.
  • Understanding how to derive expressions like $$d/dx(4x^3 - 2x + 6) = 12x^2 - 2$$ lays a solid foundation for tackling more complex functions. In real-world scenarios, derivatives represent rates of change such as velocity or profit margins, allowing for better decision-making based on trends. Mastery of these basic concepts enables one to approach calculus with confidence and apply these skills to solve practical problems involving motion, optimization, and growth rates effectively.

Related terms

Derivative: A derivative is a measure of how a function changes as its input changes, typically represented as $$f'(x)$$ or $$d/dx(f(x))$$.

Polynomial: A polynomial is a mathematical expression that consists of variables raised to whole number exponents and their coefficients, like $$4x^3 - 2x + 6$$.

Power Rule: The power rule is a basic differentiation rule stating that if you have a term $$ax^n$$, its derivative is $$nax^{n-1}$$.