Question

Find the derivative of \[{\left( {3{x^2} - 7x + 3} \right)^{5/2}}\] with respect to \[x\].

Answer 1

Hint: As the given equation is quadratic equation of the form \[a{x^2} - bx + c\], in which x is an unknown term and a function is said to be differentiable if its derivative exists and here we need to find the derivative with respect to \[x\] , hence differentiate the terms with \[\dfrac{d}{{dx}}\] of each term as given in the equation.

Complete step by step answer:
We need to find the derivative with respect to \[x\]. The equation given is of the form \[a{x^2} - bx + c\] in which to calculate the derivative of a sum, we simply take the sum of the derivatives.If we want to find the derivative of a difference, we simply find the difference of the derivatives.If we want to find the derivative of a product, we use the product rule for derivatives.As the given function is
\[f(x) = {\left( {3{x^2} - 7x + 3} \right)^{5/2}}\]

To find its derivative with respect to \[x\] let us write it in simplified manner
\[\dfrac{d}{{dx}}{\left( {3{x^2} - 7x + 3} \right)^{5/2}}\]
Further simplifying the terms, we get
\[\dfrac{5}{2}{\left( {3{x^2} - 7x + 3} \right)^{5/2 - 1}} \cdot \dfrac{d}{{dx}}\left( {3{x^2} - 7x + 3} \right)\]
Now let us expand the above terms with respect to \[x\]
\[\dfrac{{5\left( {3 \cdot \dfrac{d}{{dx}}\left[ {{x^2}} \right] - 7 \cdot \dfrac{d}{{dx}}\left[ x \right] + \dfrac{d}{{dx}}\left[ 3 \right]} \right){{\left( {3{x^2} - 7x + 3} \right)}^{3/2}}}}{2}\]

After derivative the terms with respect to \[x\] we get
\[\dfrac{{5\left( {3 \cdot 2x - 7 \cdot 1 + 0} \right){{\left( {3{x^2} - 7x + 3} \right)}^{3/2}}}}{2}\]
As we can see that the derivative of \[{x^2}\]becomes \[2x\], derivative of \[x\]is \[1\] and derivative of \[3\] is \[0\].
Simplifying the terms, with respect to \[x\] we get
\[\dfrac{{5\left( {6x - 7} \right){{\left( {3{x^2} - 7x + 3} \right)}^{3/2}}}}{2}\]
 \[\Rightarrow\dfrac{5}{2}\left( {6x - 7} \right){\left( {3{x^2} - 7x + 3} \right)^{3/2}}\]
Hence, after further simplification we get the final derivative as
\[\therefore\dfrac{{\left( {30x - 35} \right){{\left( {3{x^2} - 7x + 3} \right)}^{3/2}}}}{2}\]
Therefore, this is the final derivative term with respect to \[x\].

Note: To find any type of derivative with respect to \[x\] take \[\dfrac{d}{{dx}}\] and with respect to \[y\]take \[\dfrac{d}{{dy}}\] . It is based on the derivative terms asked in the equation.Hence here are some the rules to find any kind of derivative asked: To find the derivatives of sum
\[\left[ {f\left( x \right) + g\left( x \right)} \right]' = f'\left( x \right) + g'\left( x \right)\]
To find the derivatives of difference
\[\left[ {f\left( x \right) - g\left( x \right)} \right]' = f'\left( x \right) - g'\left( x \right)\]
To find the derivatives of product
\[\left[ {f\left( x \right) \cdot g\left( x \right)} \right]' = f'\left( x \right)g\left( x \right) + g'\left( x \right)f\left( x \right)\]