$\"\"$

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The length of ladder is $\"\"$ .

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Observe the figure, the ladder forms right angle triangle.

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Consider length of wall is $\"\"$ and base of the triangle is $\"\"$ .

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Rate of change in $\"\"$ is $\"\"$ .

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From pythagoras rule $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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From the power rule of derivative: $\"\"$ .

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$\"\"$

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$\"\"$

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Substitute $\"\"$ in the above equation.

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$\"\"$

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$\"\"$

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$\"\"$

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(a).

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Find $\"\"$ when $\"\"$ .

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Therefore

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ in the $\"\"$ .

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$\"\"$

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$\"\"$

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Find $\"\"$ when $\"\"$ .

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Therefore

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ in the $\"\"$ .

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$\"\"$

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$\"\"$

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Find $\"\"$ when $\"\"$ .

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Therefore

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ in the $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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(b).

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Find $\"\"$ ,when base of the ladder is $\"\"$ i.e $\"\"$ .

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When $\"\"$ , $\"\"$ and $\"\"$ .

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The area of triangle $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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The product rule of the derivative: $\"\"$ .

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$\"\"$

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Substitute $\"\"$ , $\"\"$ and $\"\"$ . and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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(c).

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Find $\"\"$ when $\"\"$ .

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From the triangle $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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Chain rule of derivative: $\"\"$ .

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$\"\"$

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Substitute $\"\"$ , $\"\"$ and $\"\"$ and $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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(a). $\"\"$ and $\"\"$ and $\"\"$ .

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(b). $\"\"$ .

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(c). $\"\"$ .