Chapter 2: Q 88 (page 212)

Use the chain rule twice to prove that $\frac{d}{d x} (f (u (v (x)))) = f' (u (v (x))) u' (v (x)) v' (x)$

Short Answer

Expert verified

$\frac{d}{d x} (f (u (v (x)))) = \frac{d}{d x} f (u (v (x))) \times \frac{d}{d x} u (v (x)) = f' (u (v (x))) \frac{d}{d x} u (v (x)) = f' (u (v (x))) u' (v (x)) \frac{d}{d x} v (x) \Rightarrow \frac{d}{d x} u (v (x)) = f' (u (v (x))) u' (v (x)) v' (x)$

Hence proved.

Step by step solution

Step 1. given Information

We have to prove the following derivative :-

$\frac{d}{d x} (f (u (v (x)))) = f' (u (v (x))) u' (v (x)) v' (x)$

We have to use chain rule twice to prove this derivative.

Step 2. Proof of derivative

Take the left hand side :-

$\frac{d}{d x} (f (u (v (x))))$

Use chain rule on the function $f$ :-

role="math" localid="1648908948871" $\frac{d}{d x} (f (u (v (x)))) = \frac{d}{d x} f (u (v (x))) \times \frac{d}{d x} u (v (x)) \Rightarrow \frac{d}{d x} (f (u (v (x)))) = f' (u (v (x))) \frac{d}{d x} u (v (x))$

Now use chain rule again on the function $u$ , then we have :-

role="math" localid="1648908979607" $\frac{d}{d x} u (v (x)) = f' (u (v (x))) u' (v (x)) \frac{d}{d x} v (x) \Rightarrow \frac{d}{d x} u (v (x)) = f' (u (v (x))) u' (v (x)) v' (x)$

This is equal to right hand side.

Hence proved.

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Use the chain rule twice to prove that $\frac{d}{d x} (f (u (v (x)))) = f' (u (v (x))) u' (v (x)) v' (x)$

Short Answer

Step by step solution

Step 1. given Information

Step 2. Proof of derivative

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Use the chain rule twice to prove that $\frac{d}{d x} (f (u (v (x)))) = f' (u (v (x))) u' (v (x)) v' (x)$