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How to prove that a function is differentiable? |
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Answer» Answer: example Step-by-step explanation: To show that f is DIFFERENTIABLE at all x∈R, we must show that f′(x) exists at all x∈R. Recall that f is differentiable at x if limh→0f(x+H)−f(x)h exists. So for f(x)=−5x, we examine limh→0−5(x+h)−(−5x)h=limh→0−5hh=limh→0−5=−5 And so we see that f is differentiable at all x∈R with derivative f′(x)=−5. We could also say that if g(x) and h(x) are differentiable, then so too is f(x)=g(x)h(x) and that f′(x)=g′(x)h(x)+g(x)h′(x). Then let g(x)=x,h(x)=−5, noting that both are differentiable with DERIVATES 1 and 0 respectively, LEADING to the same result. |
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