Gardiolo0j
2022-09-29
Answered

Find the derivative of $h=(x)=\frac{4{x}^{3}-7x+8}{x}$

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Colin Dougherty

Answered 2022-09-30
Author has **8** answers

Step 1

Given: $h=(x)=\frac{4{x}^{3}-7x+8}{x}$

To find- The derivative of the above function.

Identity Used- Using the derivative of the quotient function as

$\frac{d}{dx}(\frac{f(x)}{g(x)})=\frac{g(x)\times {f}^{\prime}(x)-f(x)\times {g}^{\prime}(x)}{g(x{)}^{2}}$, where ,br.f(x) and g(x) are the function of x.

Step 2

Explanation- Rewrite the given expression,

$h(x)=\frac{4{x}^{3}-7x+8}{x}$

Using the derivative of the quotient function, differentiating the above expression w.r.t. x, we get,

${h}^{\prime}(x)=\frac{x(12{x}^{2}-7)-(4{x}^{3}-7x+8)\cdot 1}{{x}^{2}}$

$=\frac{12{x}^{3}-7x-4{x}^{3}+7x-8}{{x}^{2}}$

$=\frac{8{x}^{3}-8}{{x}^{2}}$

$=\frac{8({x}^{3}-1)}{{x}^{2}}$

So, the derivative of the above function is

${h}^{\prime}(x)=\frac{8({x}^{3}-1)}{{x}^{2}}$

Answer- the derivative of the function

$\frac{4{x}^{3}-7x+8}{x}$ is ${h}^{\prime}(x)=\frac{8({x}^{3}-1)}{{x}^{2}}$

Given: $h=(x)=\frac{4{x}^{3}-7x+8}{x}$

To find- The derivative of the above function.

Identity Used- Using the derivative of the quotient function as

$\frac{d}{dx}(\frac{f(x)}{g(x)})=\frac{g(x)\times {f}^{\prime}(x)-f(x)\times {g}^{\prime}(x)}{g(x{)}^{2}}$, where ,br.f(x) and g(x) are the function of x.

Step 2

Explanation- Rewrite the given expression,

$h(x)=\frac{4{x}^{3}-7x+8}{x}$

Using the derivative of the quotient function, differentiating the above expression w.r.t. x, we get,

${h}^{\prime}(x)=\frac{x(12{x}^{2}-7)-(4{x}^{3}-7x+8)\cdot 1}{{x}^{2}}$

$=\frac{12{x}^{3}-7x-4{x}^{3}+7x-8}{{x}^{2}}$

$=\frac{8{x}^{3}-8}{{x}^{2}}$

$=\frac{8({x}^{3}-1)}{{x}^{2}}$

So, the derivative of the above function is

${h}^{\prime}(x)=\frac{8({x}^{3}-1)}{{x}^{2}}$

Answer- the derivative of the function

$\frac{4{x}^{3}-7x+8}{x}$ is ${h}^{\prime}(x)=\frac{8({x}^{3}-1)}{{x}^{2}}$

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