Math, asked by PRINCESS100001, 8 days ago

Q1.What is the derivative wrt x of y = (3x² + 5)(log x - e)​

Answers

Answered by PRINCE100001
11

Step-by-step explanation:

Solution :

\begin{gathered}\textsf{Function of y =}\:\sf{(3x^{2} + 5)[In(x) - e]} \\ \\ \end{gathered} </p><p>

\begin{gathered}:\implies \sf{\dfrac{dy}{dx} = (3x^{2} + 5)[In(x) - e]} \\ \\ \end{gathered} </p><p>

\begin{gathered}\textsf{Now, by using the quotient rule of differentiation and substituting the values of v and u} \\ \textsf{ in it, we get :} \\ \\ \end{gathered} </p><p>

</p><p>\begin{gathered}\underline{\sf{\bigstar\: Quotient\:rule\:of\: differentiation:-}} \\ \\ \sf{\dfrac{d(uv)}{dx} = (u)\cdot\dfrac{d(v)}{dx} + (v)\cdot\dfrac{d(u)}{dx}} \\ \\ \end{gathered} </p><p>

\begin{gathered}\sf{Here,} \\ \bullet\:\sf{The\:value\:of\:u = (3x^{2} + 5)} \\ \bullet\:\sf{The\:value\:of\:v = [In(x) - e]} \\ \\ \end{gathered} </p><p>

\begin{gathered}:\implies \sf{\dfrac{d(uv)}{dx} = (3x^{2} + 5)\cdot\dfrac{d[In(x) - e]}{dx} + [In(x) - e]\cdot\dfrac{d(3x^{2} + 5)}{dx}} \\ \\ \end{gathered} </p><p>

\begin{gathered}:\implies \sf{\dfrac{d(uv)}{dx} = (3x^{2} + 5)\cdot\dfrac{d[In(x)]}{dx} + \dfrac{d(-e)}{dx} + [In(x) - e]\cdot\dfrac{d(3x^{2})}{dx} + \dfrac{d(5)}{dx}} \\ \\ \end{gathered} </p><p>

\begin{gathered}\textsf{Now, by using the power rule of differentiation}\:\textsf{and constant rule of differentiation, we get :} \\ \\ \end{gathered} </p><p>

\begin{gathered}\underline{\sf{\bigstar\: Power\:rule\:of\: differentiation:-}} \\ \\ \sf{\dfrac{d(x^{n})}{dx} = n\cdot x^{(n - 1)}} \\ \\ \underline{\sf{\bigstar\: Constant\:rule\:of\: differentiation:-}} \\ \\ \sf{\dfrac{d(c)}{dx} = 0} \\ \\ \end{gathered} </p><p></p><p>

</p><p>\begin{gathered}:\implies \sf{\dfrac{d(uv)}{dx} = (3x^{2} + 5)\cdot\bigg(\dfrac{1}{x} + 0\bigg) + [In(x) - e]\cdot(6x + 0)} \\ \\ \end{gathered} </p><p>

\begin{gathered}:\implies \sf{\dfrac{d(uv)}{dx} = \dfrac{(3x^{2} + 5)}{x} + 6x[In(x) - e]} \\ \\ \end{gathered} </p><p>

\begin{gathered}\boxed{\therefore \sf{\dfrac{dy}{dx} = \dfrac{(3x^{2} + 5)}{x} + 6x[In(x) - e]}} \\ \\ \end{gathered} </p><p>

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