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Chapter 12: Q. 1 TB (page 963)

Chain rule: If $$f$$ is a function of $$x$$ and $$x$$ is a function of $$t$$, how is the chain rule used to find the rate of change of $$f$$ with respect to $$t$$?

Short Answer

Expert verified

The rate of change of $$f$$ with respect to $$t$$ found using the Chain's rule is $$\frac{df}{dt}=\frac{df}{dx} \times \frac{dx}{dt}$$

Step by step solution

01

Step 1. Given Information

$$f$$ is a function of $$x$$ and $$x$$ is a function of $$t$$

02

Step 2. Explanation

It is given that, $$f$$ is a function of $$x$$ and $$x$$ is a function of $$t$$

The rate of change of $$f$$ with respect to $$t$$ is $$\frac{df}{dt}$$

We can solve $$\frac{df}{dt}$$ using the Leibniz's notation of the Chain's rule.

Hence, using Chain's rule, we get the rate of change of $$f$$ with respect to $$t$$ as $$\frac{df}{dt}=\frac{df}{dx} \times \frac{dx}{dt}$$

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