Question

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

Answer 1

Hint: To find the derivative, consider the function as Y, and differentiate the function with respect to x. The exponent will come down and the remaining function will be differentiated with respect to x.

Complete step-by-step answer:
If f(x) = h(g(x)) then f (x) = h (g(x)) × g (x). In words: differentiate the 'outside' function, and then multiply by the derivative of the 'inside' function.
The formula for derivative of \[{{x}^{n}}=n\cdot {{x}^{n-1}}\].
First step let the given expression be Y
Differentiate both sides with respect to x.
\[\begin{align}
  & Y={{\left( 3{{x}^{2}}-7x+3 \right)}^{\dfrac{5}{2}}} \\
 & \dfrac{dY}{dx}=\dfrac{5}{2}\cdot {{\left( 3{{x}^{2}}-7x+3 \right)}^{\dfrac{5}{2}-1}}\cdot \left[ 3\dfrac{d}{dx}\left( {{x}^{2}} \right)-7\dfrac{d}{dx}\left( x \right)+\dfrac{d}{dx}\left( 3 \right) \right] \\
 & \dfrac{dY}{dx}=\dfrac{5}{2}\cdot {{\left( 3{{x}^{2}}-7x+3 \right)}^{\dfrac{3}{2}}}\cdot \left[ 3\left( 2\cdot {{x}^{2-1}} \right)-7\left( 1 \right)+\left( 0 \right) \right] \\
 & \dfrac{dY}{dx}=\dfrac{5}{2}\cdot {{\left( 3{{x}^{2}}-7x+3 \right)}^{\dfrac{3}{2}}}\cdot \left[ 3\left( 2\cdot x \right)-7\cdot 1 \right] \\
 & \dfrac{dY}{dx}=\dfrac{5}{2}\cdot {{\left( 3{{x}^{2}}-7x+3 \right)}^{\dfrac{3}{2}}}\cdot \left( 6x-7 \right) \\
 & \\
\end{align}\]
Thus, the derivative of \[{{\left( 3{{x}^{2}}-7x+3 \right)}^{\dfrac{5}{2}}}\] with respect to x is \[\dfrac{5}{2}\cdot {{\left( 3{{x}^{2}}-7x+3 \right)}^{\dfrac{3}{2}}}\cdot \left( 6x-7 \right)\].

Note: The process of finding a derivative is called differentiation. The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change of x. Derivatives are a fundamental tool of calculus.