Question

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

Answer 1

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
$\left[ {f\left( x \right) \cdot g\left( x \right)} \right]' = f'\left( x \right)g\left( x \right) + g'\left( x \right)f\left( x \right)$