Find the Derivative - d/d@VAR f(x)=8x^4
The given problem is asking to perform a mathematical operation called differentiation on the function f(x) = 8x^4, with respect to the variable x. Differentiation is a fundamental concept in calculus that involves finding the derivative, which represents the rate at which the function's value changes as the variable changes. Here, '@VAR' presumably stands for a typo or placeholder and should simply be denoted as 'x'. The derivative of a function gives the slope of the tangent line to the graph of the function at any point. The operation indicated by d/dx is instructing to differentiate the function with respect to x to find its derivative.
$f \left(\right. x \left.\right) = 8 x^{4}$
Answer
Solution:
Step 1:
Identify that the number $8$ is a constant factor in the expression $8x^4$ and can be pulled out of the derivative operation. Thus, we express the derivative as $8 \cdot \frac{d}{dx}(x^4)$.
Step 2:
Apply the Power Rule for differentiation, which tells us that the derivative of $x^n$ with respect to $x$ is $n \cdot x^{n-1}$. In this case, with $n=4$, we get $8 \cdot (4x^{4-1})$.
Step 3:
Simplify the expression by multiplying the constants $8$ and $4$ to obtain the final derivative, which is $32x^3$.
Knowledge Notes:
The problem involves finding the derivative of a function with respect to a variable, which is a fundamental concept in calculus. The derivative represents the rate at which the function's value changes with respect to changes in the variable.
Constant Multiplication Rule: When taking the derivative of a constant multiplied by a function, the constant can be factored out and the derivative of the function is taken normally. Mathematically, if $c$ is a constant and $f(x)$ is a function, then $\frac{d}{dx}[cf(x)] = c \cdot \frac{d}{dx}[f(x)]$.
Power Rule: This is a basic rule for differentiating expressions of the form $x^n$ where $n$ is any real number. The rule states that $\frac{d}{dx}[x^n] = n \cdot x^{n-1}$. This rule is applied directly to find the derivative of monomials.
Simplification: After applying the differentiation rules, it's common to simplify the expression by combining like terms or multiplying constants together, as seen in the final step of the solution.
In LaTeX format, the derivative of $f(x) = 8x^4$ is expressed as $\frac{d}{dx}[8x^4]$. The solution involves applying the constant multiplication rule and the power rule, resulting in $32x^3$.
