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Chapter 4: Q. 79 (page 363)

$T h e functions F, G are differentiable also G (x) = F (x) + C . Where, Cis any constant . P rove that F' (x) = G (x) .$

Short Answer

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Step by step solution

01

Step 1. To prove

$F' (x) = G (x) .$

02

Step 2. Proving

$N o w, \frac{d}{d x} (G (x)) = \frac{d}{d x} (F (x) + C) G' (x) = F' (x) + 0 G' (x) = F' (x) Therefore, it is proved that F' (x) = G' (x)$

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Most popular questions from this chapter

Fill in each of the blanks:

(a) $\int d x = x^{6} + C .$

(b) $x^{6}$ is an antiderivative of .

(c) The derivative of $x^{6}$ is .

Determine which of the limit of sums in Exercises 47–52 are infinite and which are finite. For each limit of sums that is finite, compute its value

$\lim_{n \to \infty} \sum_{k = 1}^{n} \frac{{(k + 1)}^{2}}{n^{3} - 1}$

Find the sum or quantity without completely expanding or calculating any sums.

Given $\sum_{k = 3}^{10} a_{k} = 12$ and $\sum_{k = 2}^{10} a_{k} = 23$ , find $a_{2}$ .

Given a simple proof that $\sum_{k = 5}^{n} (a_{k} + b_{k}) = \sum_{k = 5}^{n} a_{k} + \sum_{k = 5}^{n} b_{k}$

Find the sum or quantity without completely expanding or calculating any sums.

Given $\sum_{k = 0}^{25} k = 325$ and $\sum_{k = 3}^{28} {(k - 3)}^{2} = 14, 910$ , find the value of $\sum_{k = 3}^{25} (k^{2} - 5 k + 9)$ .

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