Question

If \(F\left( x \right) = f\left( {xf\left( {xf\left( x \right)} \right)} \right)\), where \(f\left( 1 \right) = 2\), \(f\left( 2 \right) = 3\), \(f'\left( 1 \right) = 4,{\rm{ }}f'\left( 2 \right) = 5\), and \(f'\left( 3 \right) = 6\), find \(F'\left( 1 \right)\).

Short answer

The value of \(F'\left( 1 \right)\) is 198.

Step-by-step solution

The Chain Rule

The composite function \(F = f \circ g\), or \(F\left( x \right) = f\left( {g\left( x \right)} \right)\) is differentiableat \(x\) and \(F'\) is obtained as:

\(F'\left( x \right) = f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right)\)

In Leibniz notation, the derivative is shown below:

\(\frac{{dy}}{{dx}} = \frac{{dy}}{{du}}\frac{{du}}{{dx}}\)

Determine the value of \(F'\left( 1 \right)\)

Use the product rule and the chain rule to differentiate the function \(F\left( x \right)\) as shown below:

\(\begin{aligned}F'\left( x \right) &= \frac{d}{{dx}}f\left( {xf\left( {xf\left( x \right)} \right)} \right)\\ &= f'\left( {xf\left( {xf\left( x \right)} \right)} \right)\frac{d}{{dx}}\left( {xf\left( {xf\left( x \right)} \right)} \right)\\ &= f'\left( {xf\left( {xf\left( x \right)} \right)} \right) \cdot \left( {x \cdot \frac{d}{{dx}}\left( {f\left( {xf\left( x \right)} \right)} \right) + f\left( {xf\left( x \right)} \right)\frac{d}{{dx}}x} \right)\\ &= f'\left( {xf\left( {xf\left( x \right)} \right)} \right) \cdot \left( {x\left( {f'\left( {xf\left( x \right)} \right)} \right) \cdot \frac{d}{{dx}}\left( {xf\left( x \right)} \right) + f\left( {xf\left( x \right)} \right) \cdot 1} \right)\\ &= f'\left( {xf\left( {xf\left( x \right)} \right)} \right) \cdot \left( {x\left( {f'\left( {xf\left( x \right)} \right)} \right) \cdot \left( {x\frac{d}{{dx}}f\left( x \right) + f\left( x \right)\frac{d}{{dx}}x} \right) + f\left( {xf\left( x \right)} \right)} \right)\\ &= f'\left( {xf\left( {xf\left( x \right)} \right)} \right) \cdot \left( {x\left( {f'\left( {xf\left( x \right)} \right)} \right) \cdot \left( {xf'\left( x \right) + f\left( x \right) \cdot 1} \right) + f\left( {xf\left( x \right)} \right)} \right)\end{aligned}\)

Use the given values to determine the value of \(F'\left( 1 \right)\) as shown below:

\(\begin{aligned}F'\left( 1 \right) &= f'\left( {f\left( {f\left( 1 \right)} \right)} \right) \cdot \left( {\left( {f'\left( {f\left( 1 \right)} \right)} \right) \cdot \left( {f'\left( 1 \right) + f\left( 1 \right) \cdot 1} \right) + f\left( {f\left( 1 \right)} \right)} \right)\\ &= f'\left( {f\left( 2 \right)} \right) \cdot \left( {\left( {f'\left( 2 \right)} \right) \cdot \left( {4 + 2} \right) + f\left( 2 \right)} \right)\\ &= f'\left( 3 \right) \cdot \left( {5 \cdot 6 + 3} \right)\\ &= 6 \cdot 33\\ &= 198\end{aligned}\)

Thus, the value of \(F'\left( 1 \right)\) is 198.