Question Statement
Find the first and second derivatives of the function:
f(x)=30x2−x3
Background and Explanation
This problem requires applying the power rule for differentiation to find successive derivatives of a polynomial function. The power rule states that dxd(xn)=nxn−1, which we will use along with the constant multiple rule to differentiate each term systematically.
Solution
We start by differentiating f(x)=30x2−x3 with respect to x. Using the difference rule and constant multiple rule, we can differentiate each term separately:
f′(x)=30dxd(x2)−dxd(x3)
Applying the power rule dxd(xn)=nxn−1 to each term:
f′(x)=30(2x)−3x2=60x−3x2
Thus, the first derivative is:
f′(x)=60x−3x2
To find the second derivative, we differentiate f′(x) with respect to x:
f′′(x)=dxd(60x−3x2)
Again applying the constant multiple rule and power rule:
f′′(x)=60dxd(x)−3dxd(x2)=60(1)−3(2x)=60−6x
Therefore, the second derivative is:
f′′(x)=60−6x
- Power Rule: dxd(xn)=nxn−1
- Constant Multiple Rule: dxd[c⋅f(x)]=c⋅f′(x)
- Difference Rule: dxd[f(x)−g(x)]=f′(x)−g′(x)
- Derivative of x: dxd(x)=1
Summary of Steps
- Identify the original function: f(x)=30x2−x3
- Calculate the first derivative f′(x):
- Differentiate 30x2 to get 60x
- Differentiate −x3 to get −3x2
- Combine: f′(x)=60x−3x2
- Calculate the second derivative f′′(x):
- Differentiate 60x to get 60
- Differentiate −3x2 to get −6x
- Combine: f′′(x)=60−6x