Home fsc 1st year math solutions Derivative of a function example 01

Derivative of a function example 01

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Derivative of a function
Derivative of a function

Derivative of a function example 01

\textbf{Question:} Find the derivative of the function f(x) = \arcsin(3x^2 + 2x). \textbf{Solution:}

\textbf{Step 1:} Start by differentiating f(x) with respect to x:

    \[\frac{d}{dx} \left( \arcsin(3x^2 + 2x) \right)\]

\textbf{Step 2:} Apply the chain rule. Let u = 3x^2 + 2x, then \frac{du}{dx} = 6x + 2.

    \[\frac{d}{du} \left( \arcsin(u) \right) \cdot \frac{du}{dx}\]

\textbf{Step 3:} Recall that the derivative of \arcsin(u) is \frac{1}{\sqrt{1 - u^2}}.

    \[\frac{1}{\sqrt{1 - u^2}} \cdot (6x + 2)\]

\textbf{Step 4:} Substitute back u = 3x^2 + 2x.

    \[\frac{6x + 2}{\sqrt{1 - (3x^2 + 2x)^2}}\]

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Thus, the derivative of f(x) is:

    \[\frac{6x + 2}{\sqrt{1 - (3x^2 + 2x)^2}}\]

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