2nd Derivatives

#051 / Math Medium

Calculate the second derivative of (x^7)-(x^3)+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:

$$$\frac{d^2}{dx^2}(x^7-x^3+2)=\frac{d^2}{dx^2}(x^7)-\frac{d^2}{dx^2}(x^3)+\frac{d^2}{dx^2}(2)$$$

Applying the 2nd power rule to the first section of the derivative with $$n=7$$ :

$$$\frac{d^2}{dx^2}[x^n]=n(n-1)(x^{n-2})\rightarrow\frac{d^2}{dx^2}(x^7)=42x^5$$$

Applying the 2nd power rule to the second section of the derivative with $$n=3$$ :

$$$\frac{d^2}{dx^2}[x^n]=n(n-1)(x^{n-2})\rightarrow\frac{d^2}{dx^2}(x^3)=6x$$$

Applying the constant rule to the third section of the derivative:

$$$\frac{d^2}{dx^2}[2x^0]=0$$$

Finally,

$$$\frac{d^{2}}{dx^{2}}(x^{7})-\frac{d^{2}}{dx^{2}}(x^{3})+\frac{d^{2}}{dx^{2}}(2)=42x^5-6x$$$

$$$42x^5-6x$$$

Time Analysis

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