Question Statement

Find the fourth derivative $\frac{d^{4}y}{d{x}^4}$ of the function:

$y=4x^7+x^6-x^4$

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Background and Explanation

This problem involves computing higher-order derivatives (successive derivatives beyond the first) of a polynomial function. You will need to apply the power rule for differentiation, which states that $\frac{d}{dx}(x^n)=n{x}^{n-1}$, repeatedly four times while carefully tracking the coefficients at each step.

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Solution

We begin with the given function:

$y=4x^7+x^6-x^4$

First Derivative

Differentiate each term with respect to $x$ using the power rule $\frac{d}{dx}(x^n)=n{x}^{n-1}$ and the constant multiple rule:

$\begin{array}{rl}\frac{dy}{dx}& =4\frac{d}{dx}\left(x^7\right)+\frac{d}{dx}\left(x^6\right)-\frac{d}{dx}\left(x^4\right)\\ & =4\left(7x^6\right)+6x^5-4x^3\\ & =28x^6+6x^5-4x^3\end{array}$

Second Derivative

Differentiate $\frac{dy}{dx}$ again with respect to $x$:

$\begin{array}{rl}\frac{d^{2}y}{d{x}^2}& =28\frac{d}{dx}(x^6)+6\frac{d}{dx}(x^5)-4\frac{d}{dx}(x^3)\\ & =28\left(6x^5\right)+6\left(5x^4\right)-4\left(3x^2\right)\\ & =168x^5+30x^4-12x^2\end{array}$

Third Derivative

Differentiate $\frac{d^{2}y}{d{x}^2}$ once more with respect to $x$:

$\begin{array}{rl}\frac{d^{3}y}{d{x}^3}& =168\frac{d}{dx}(x^5)+30\frac{d}{dx}(x^4)-12\frac{d}{dx}(x^2)\\ & =168\left(5x^4\right)+30\left(4x^3\right)-12(2x)\\ & =840x^4+120x^3-24x\end{array}$

Fourth Derivative

Finally, differentiate $\frac{d^{3}y}{d{x}^3}$ to obtain the required fourth derivative. Note that the derivative of $x$ is $1$:

$\begin{array}{rl}\frac{d^{4}y}{d{x}^4}& =840\frac{d}{dx}(x^4)+120\frac{d}{dx}(x^3)-24\frac{d}{dx}(x)\\ & =840\left(4x^3\right)+120\left(3x^2\right)-24(1)\\ & =3360x^3+360x^2-24\end{array}$

Therefore, the fourth derivative is:

$\frac{d^{4}y}{d{x}^4}=3360x^3+360x^2-24$

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Key Formulas or Methods Used

• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$
• Constant Multiple Rule: $\frac{d}{dx}[c\cdot f(x)]=c\cdot f^{\prime }(x)$
• Sum/Difference Rule: $\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime }(x)\pm g^{\prime }(x)$
• Higher-Order Derivative Notation: $\frac{d^{4}y}{d{x}^4}=\frac{d}{dx}\left(\frac{d^{3}y}{d{x}^3}\right)$ (successive differentiation)

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Summary of Steps

1. First differentiation: Apply the power rule to each term of $y=4x^7+x^6-x^4$ to obtain $\frac{dy}{dx}=28x^6+6x^5-4x^3$
2. Second differentiation: Differentiate the first derivative to get $\frac{d^{2}y}{d{x}^2}=168x^5+30x^4-12x^2$
3. Third differentiation: Differentiate the second derivative to get $\frac{d^{3}y}{d{x}^3}=840x^4+120x^3-24x$
4. Fourth differentiation: Differentiate the third derivative, remembering that $\frac{d}{dx}(x)=1$, to obtain the final answer $\frac{d^{4}y}{d{x}^4}=3360x^3+360x^2-24$