# Find the first and second derivatives of the given function. f(x)=5x^{3}-6x^{2}+6

Find the first and second derivatives of the given function.
$f\left(x\right)=5{x}^{3}-6{x}^{2}+6$
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Step 1
The given function is $f\left(x\right)=5{x}^{3}-6{x}^{2}+6$.
The derivative of ${x}^{n}$ is given by: $\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}$
and derivative of constant is zero, that is $\frac{d}{dx}\left(c\right)=0$.
Step 2
To find the first derivative of the given function $f\left(x\right)=5{x}^{3}-6{x}^{2}+6$, differentiate this function with respect to x.
$f\left(x\right)=5{x}^{3}-6{x}^{2}+6$
${f}^{\prime }\left(x\right)=\frac{d}{dx}\left(6{x}^{3}-6{x}^{2}+6\right)$ (since, $\frac{d}{dx}f\left(x\right)={f}^{\prime }\left(x\right)$)
${f}^{\prime }\left(x\right)=\frac{d}{dx}\left(5{x}^{3}\right)+\frac{d}{dx}\left(-6{x}^{2}\right)+\frac{d}{dx}\left(6\right)$
${f}^{\prime }\left(x\right)=5\frac{d}{dx}\left({x}^{3}\right)-6\frac{d}{dx}\left({x}^{2}\right)+\frac{d}{dx}\left(6\right)$
${f}^{\prime }\left(x\right)=5\left(3{x}^{\left(3-1\right)}\right)-6\left(2{x}^{\left(2-1\right)}\right)+0$
${f}^{\prime }\left(x\right)=15{x}^{2}-12x$
Therefore, first derivative of the given function is ${f}^{\prime }\left(x\right)=15{x}^{2}-12x$.
Step 3
To find second derivative of the given function, differentiate first derivative ${f}^{\prime }\left(x\right)=15{x}^{2}-12x$ with respect to x.
$f{}^{″}\left(x\right)=\frac{d}{dx}\left(15{x}^{2}-12x\right)$ (since, $\frac{{d}^{2}}{{dx}^{2}}f\left(x\right)=f{}^{″}\left(x\right)$)
chumants6g
$\frac{{d}^{2}}{{dx}^{2}}\left(5{x}^{3}-6{x}^{2}+6\right)$
$\frac{d}{dx}\left(5{x}^{3}-6{x}^{2}+6\right)=15{x}^{2}-12x$
$=\frac{d}{dx}\left(15{x}^{2}-12x\right)$
$\frac{d}{dx}\left(15{x}^{2}-12x\right)=30x-12$
=30x-12