Derivatives
Calculate the second derivative of (x^8)-(x^2)+2
Expand Hint
The power rule for the first derivative:
$$$\frac{d}{dx}[x^n]=n\cdot x^{n-1}$$$
Hint 2
Using the power rule for the first derivative, and applying it twice, we’ll get the second derivative power rule:
$$$\frac{d^2}{dx^2}[x^n]=\frac{d}{dx}\frac{d}{dx}[x^n]=\frac{d}{dx}[nx^{n-1}]=n\frac{d}{dx}[x^{n-1}]=n(n-1)(x^{n-2})$$$
Using the power rule for the first derivative, and applying it twice, we'll get the second derivative power rule:
$$$\frac{d^2}{dx^2}[x^n]=\frac{d}{dx}\frac{d}{dx}[x^n]=\frac{d}{dx}[nx^{n-1}]=n\frac{d}{dx}[x^{n-1}]=n(n-1)(x^{n-2})$$$
This calculus problem can be written as:
$$$\frac{d^2}{dx^2}(x^8-x^2+2)=\frac{d^2}{dx^2}(x^8)-\frac{d^2}{dx^2}(x^2)+\frac{d^2}{dx^2}(2)$$$
Applying the 2nd power rule to the first section of the derivative with
$$n=8$$
:
$$$\frac{d^2}{dx^2}[x^n]=n(n-1)(x^{n-2})\rightarrow\frac{d^2}{dx^2}(x^8)=56x^6$$$
Applying the 2nd power rule to the second section of the derivative with
$$n=2$$
:
$$$\frac{d^2}{dx^2}[x^n]=n(n-1)(x^{n-2})\rightarrow\frac{d^2}{dx^2}(x^2)=2$$$
Applying the constant rule to the third section of the derivative:
$$$\frac{d^2}{dx^2}[2x^0]=0$$$
Finally,
$$$\frac{d^{2}}{dx^{2}}(x^{8})-\frac{d^{2}}{dx^{2}}(x^{2})+\frac{d^{2}}{dx^{2}}(2)=56x^6-2$$$
$$$56x^6-2$$$
Time Analysis
See how quickly you looked at the hint, solution, and answer. This is important for making sure you will finish the FE Exam in time.- Hint: Not clicked
- Solution: Not clicked
- Answer: Not clicked
Similar Problems from FE Sub Section: The Derivative, Derivatives
048. Derivative
051. 2nd Derivatives
122. Roller Coasters
158. A Train's Velocity
312. The Derivative
394. Acceleration
412. Hill Bomb
650. Differentiate
Similar Problems from FE Section: Differential Calculus
040. Integrals
043. A Calc Question
044. Definite Integrals
048. Derivative
051. 2nd Derivatives
077. Area Under a Curve
122. Roller Coasters
158. A Train's Velocity
308. Definite Integral
312. The Derivative
342. Inflection Point
353. Inflections
355. Multi Roots
394. Acceleration
408. Triple Integrals
412. Hill Bomb
416. Trip Integral
430. A Curve’s Area
643. Integral Finding
650. Differentiate