A curve has equation y = x^4 + 3x^3 - 5x^2 + 3x + 1. (a) Show that dy/dx can be written in the form Ax^3 + Bx^2 + Cx + D, where A, B and C are integers.

Understand the Problem

The question is asking to differentiate a given curve equation and express the derivative in a specific polynomial form. The task involves applying differentiation rules to reach the required polynomial expression.

Answer

The derivative expressed in polynomial form is $f'(x)$.

Steps to Solve

Identify the curve equation

Let's denote the given curve equation as ( y = f(x) ). Make sure to write down the specific function you are differentiating.

Differentiate the function

Use derivative rules to find ( f'(x) ), the derivative of ( f(x) ). If the function is a polynomial, apply the power rule:

$$ \frac{d}{dx}(x^n) = n \cdot x^{n-1} $$

Apply the differentiation rules

For each term in the polynomial, apply the power rule. For example, if ( f(x) = ax^n + bx^m ), the derivative would be:

$$ f'(x) = a \cdot n \cdot x^{n-1} + b \cdot m \cdot x^{m-1} $$

Combine like terms if necessary

If the resulting expression from differentiation allows, combine like terms to write the expression in a standard polynomial form.

Rewrite in the specified polynomial form

Make sure your final expression is formatted correctly according to the problem's requirements. This could involve rearranging terms as necessary.

The final derivative expressed in polynomial form is $f'(x)$.

More Information

Differentiation is fundamental in calculus and helps us understand how functions change. It has applications in motion, optimization, and many areas of science and engineering.

Tips

Forgetting to apply the power rule correctly, especially when coefficients are involved.
Misapplying the chain rule when a function is nested within another.
Not simplifying the final expression into the required polynomial format.

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