AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
How to Find the Derivative of $f(x)=5(x^3 - 2)$
Mathematics
Grade 11 (Senior High School)
Question Content
Find the derivative of the function $f(x)=5(x^3 - 2)$
Correct Answer
$f'(x)=15x^2$
Detailed Solution Steps
1
Step 1: Apply the constant multiple rule of differentiation: If $f(x)=k\\cdot g(x)$, then $f'(x)=k\\cdot g'(x)$, where $k=5$ and $g(x)=x^3-2$.
2
Step 2: Differentiate $g(x)=x^3-2$ using the power rule ($\\frac{d}{dx}[x^n]=nx^{n-1}$) and constant rule ($\\frac{d}{dx}[c]=0$): $g'(x)=3x^2 - 0=3x^2$.
3
Step 3: Multiply by the constant 5: $f'(x)=5\\cdot3x^2=15x^2$.
Knowledge Points Involved
1
Constant Multiple Rule of Differentiation
This rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function, expressed as $\\frac{d}{dx}[k\\cdot f(x)]=k\\cdot f'(x)$. It is used to simplify differentiation when a function is scaled by a constant factor.
2
Power Rule of Differentiation
The power rule defines the derivative of a monomial $x^n$ (where $n$ is a real number) as $\\frac{d}{dx}[x^n]=nx^{n-1}$. It is the foundational rule for differentiating polynomial terms.
3
Constant Rule of Differentiation
This rule states that the derivative of a constant value $c$ is 0, since a constant function has a slope of 0 at all points, expressed as $\\frac{d}{dx}[c]=0$.
Loading solution...
