Question

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

Answer 1

Hint: This type of question can be solved by using basic differentiation formulas. Here for a given function ${(3{x^2} - 7x + 3)^{\dfrac{5}{2}}}$, use the chain rule of differential calculus for finding its derivative. Chain rule of differential calculus is given by \[\dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \cdot \dfrac{{du}}{{dx}}\]. Let’s assume $u = 3{x^2} - 7x + 3$. Use the power rule formula $\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}$ for finding derivation. Simplify it to get the final derivative of the given function ${(3{x^2} - 7x + 3)^{\dfrac{5}{2}}}$.

Complete step-by-step answer:
Here the given function is ${(3{x^2} - 7x + 3)^{\dfrac{5}{2}}}$.
Let’s say function $y = {(3{x^2} - 7x + 3)^{\dfrac{5}{2}}}$.
Derivatives of such function can be obtained by using the chain rule of differential calculus.
Chain rule of differential calculus is given by \[\dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \cdot \dfrac{{du}}{{dx}}\].
Now for given function $y = {(3{x^2} - 7x + 3)^{\dfrac{5}{2}}}$, let’s assume $u = 3{x^2} - 7x + 3$.
So, $y = {(u)^{\dfrac{5}{2}}}$
Taking derivation of function y with respect x on both side of equation,
$\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}({u^{\dfrac{5}{2}}})$
Now using chain rule of differential calculus,
$\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}({u^{\dfrac{5}{2}}}) = \dfrac{d}{{du}}({u^{\dfrac{5}{2}}}) \cdot \dfrac{{du}}{{dx}}$
Using basic derivation formula, $\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}$
So, $\dfrac{d}{{du}}({u^{\dfrac{5}{2}}}) = \dfrac{5}{2} \cdot {u^{(\dfrac{5}{2} - 1)}}$
So, $\dfrac{d}{{du}}({u^{\dfrac{5}{2}}}) = \dfrac{5}{2} \cdot {u^{\dfrac{3}{2}}}$
Put equation of u, as $u = 3{x^2} - 7x + 3$,
So, $\dfrac{d}{{du}}({u^{\dfrac{5}{2}}}) = \dfrac{5}{2} \cdot {(3{x^2} - 7x + 3)^{\dfrac{3}{2}}}$
And for $\dfrac{{du}}{{dx}}$,
$\dfrac{{du}}{{dx}} = \dfrac{d}{{dx}}(3{x^2} - 7x + 3)$
Using sum of derivatives rule, \[\dfrac{d}{{dx}}(f(x) + g(x)) = \dfrac{d}{{dx}}(f(x)) + \dfrac{d}{{dx}}(g(x))\]
So, $\dfrac{{du}}{{dx}} = \dfrac{d}{{dx}}(3{x^2}) + \dfrac{d}{{dx}}( - 7x) + \dfrac{d}{{dx}}(3)$
$\dfrac{{du}}{{dx}} = \dfrac{d}{{dx}}(3{x^2}) + \dfrac{d}{{dx}}( - 7x) + \dfrac{d}{{dx}}(3)$
As 3 and -7 are constant terms with respect to x, so taking the constant term out of derivation, as $\dfrac{d}{{dx}}(a \cdot f(x)) = a \cdot \dfrac{d}{{dx}}(f(x))$,
So, \[\dfrac{{du}}{{dx}} = 3 \cdot \dfrac{d}{{dx}}({x^2}) - 7 \cdot \dfrac{d}{{dx}}(x) + 3 \cdot \dfrac{d}{{dx}}(1)\]
As we know that derivation of any constant term is zero, means $\dfrac{d}{{dx}}(a) = 0$
And using the basic derivation formula, $\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}$
So, \[\dfrac{{du}}{{dx}} = 3 \cdot 2{x^{2 - 1}} - 7 \cdot {x^{1 - 1}} + 3 \cdot (0)\]
Simplifying the above terms,
\[\dfrac{{du}}{{dx}} = 6x - 7 \cdot {x^0} + 0\]
So, \[\dfrac{{du}}{{dx}} = 6x - 7 \cdot (1)\]
So, \[\dfrac{{du}}{{dx}} = 6x - 7\]
Now putting the value of $\dfrac{d}{{du}}({u^{\dfrac{5}{2}}})$ and \[\dfrac{{du}}{{dx}}\] in $\dfrac{{dy}}{{dx}} = \dfrac{d}{{du}}({u^{\dfrac{5}{2}}}) \cdot \dfrac{{du}}{{dx}}$equation,
So, $\dfrac{{dy}}{{dx}} = \dfrac{5}{2} \cdot {(3{x^2} - 7x + 3)^{\dfrac{3}{2}}} \cdot (6x - 7)$
Arranging the terms, $\dfrac{{dy}}{{dx}} = \dfrac{5}{2} \cdot (6x - 7) \cdot {(3{x^2} - 7x + 3)^{\dfrac{3}{2}}}$.

So the derivative of the function ${(3{x^2} - 7x + 3)^{\dfrac{5}{2}}}$ with respect to x is given by $\dfrac{{dy}}{{dx}} = \dfrac{5}{2} \cdot (6x - 7) \cdot {(3{x^2} - 7x + 3)^{\dfrac{3}{2}}}$.

Note: If the given function is \[{(3{x^2} - 7x + 3)^x}\], then it is neither a power function form $({x^n})$ nor a exponential functional form $({a^x})$. So formulas for differentiation of these forms cannot be used. To derivative of such function \[y = {(3{x^2} - 7x + 3)^x}\], take natural logarithm on both sides of the equation.
So, \[\ln (y) = \ln ({(3{x^2} - 7x + 3)^x})\].
Use properties of logarithmic functions to expand term of right side,
Taking derivative of above equation on both side with respect to x,
Use chain rule of derivation, \[\dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \cdot \dfrac{{du}}{{dx}}\], product rule of derivation $\dfrac{d}{{dx}}(f(x) \cdot g(x)) = (\dfrac{d}{{dx}}f(x)) \cdot g(x) + f(x) \cdot (\dfrac{d}{{dx}}g(x))$ and some basic formula of derivation as used in above solution and simplify the solution to find derivative of function \[{(3{x^2} - 7x + 3)^x}\].