Two students attempt to prove that

MATHEMATICS · Single correct · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

Two students attempt to prove that ddx[f(x)+g(x)]=f′(x)+g′(x)\dfrac{d}{dx}[f(x)+g(x)]=f'(x)+g'(x). Student A writes: f(x+h)+g(x+h)−f(x)−g(x)h=f(x+h)−f(x)h+g(x+h)−g(x)h,\frac{f(x+h)+g(x+h)-f(x)-g(x)}{h}=\frac{f(x+h)-f(x)}{h}+\frac{g(x+h)-g(x)}{h}, and then takes the limit h→0h\to0. Student B writes: f(x+h)+g(x)−f(x)−g(x+h)h=f(x+h)−f(x)h−g(x+h)−g(x)h,\frac{f(x+h)+g(x)-f(x)-g(x+h)}{h}=\frac{f(x+h)-f(x)}{h}-\frac{g(x+h)-g(x)}{h}, and concludes f′(x)−g′(x)f'(x)-g'(x). Which statement about their work is correct? (a) Only Student B's proof is correct. (b) Both students' proofs are correct. (c) Only Student A's proof is correct. (d) Neither student's proof is correct.

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