Question

h(x)=4(x+4)^2+6$

Short answer

\(h'(x) = 8x + 32\)

Step-by-step solution

Simplify the function

The first step is to simplify the function \(h(x) = 4(x+4)^2+6\). The expression \((x+4)^2\) can be expanded as \(x^2 + 8x + 16\). We will distribute the 4 and get \(4x^2 + 32x + 64\). Adding 6 gives us \(4x^2 + 32x + 70\).

Derive h(x)

Next, we derive \(h(x) = 4x^2 + 32x + 70\). Using power rule, the derivative of \(x^n\) is \(n*x^{n-1}\). So the derivative of \(4x^2\) is \(8x\). The derivative of \(32x\) is \(32\). The derivative of a constant such as 70 is 0. Thus, \(h'(x)=8x + 32\).

Simplify the derivative

Finally, we simplify the derivative if necessary. In this case, the derivative \(h'(x) = 8x + 32\) is already in its simplest form.

Key concepts

Understanding Polynomials

Polynomials are mathematical expressions made up of variables raised to whole number powers and multiplied by coefficients. They are frequently used in calculus and algebra to model more complex underlying relationships. For example, in the expression given in the exercise, \(4(x+4)^2 + 6\), the polynomial components are evident when it is expanded.
This expansion process shows us the terms of the polynomial: \(4x^2 + 32x + 70\).

• The variables in polynomials are expressed in powers, made evident in terms like \(x^2\).
• Each term in a polynomial is usually a combination of coefficients and variable powers, such as the \(4x^2\) term.
• Polynomials may also include constant terms, like \(70\) in this case, which do not change with variations in x.

Recognizing the structure of polynomials can help to simplify and solve mathematical problems.

Simplification

Simplification is the process of making an expression more manageable by combining like terms or applying mathematical rules. This step is crucial because it helps in understanding and solving the problem more easily.
For the original function, the expression \( (x+4)^2 \) is expanded as \( x^2 + 8x + 16 \). By distributing the 4 across the expanded terms, we transform it into \( 4x^2 + 32x + 64 \).

• First, expand expressions within parentheses to reveal all terms.
• Use distribution to apply coefficients to each term of the expanded expression.
• Combine all like terms to achieve the simplest form.

Finally, by adding the constant, we find the simplified polynomial \( 4x^2 + 32x + 70 \). This simplification process is fundamental to then deriving the function.

Applying the Power Rule

The power rule is a basic and efficient technique used to find the derivative of polynomial functions quickly. It states that if you have a term \( x^n \), its derivative will be \( nx^{n-1} \). Applying this formula allows us to derive each term in the function.
For example:

• The derivative of \(4x^2\) is calculated as \(8x\), where 4 is multiplied by 2 and the power of x is reduced by 1.
• The term \(32x\) simplifies to \(32\) as the derivative of \(x\) is 1.
• Lastly, constants like \(70\) become 0, as they do not change with respect to x.

Therefore, the derivative is \(h'(x) = 8x + 32\).
The power rule helps in taking derivatives quickly, making it an indispensable tool in calculus.