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Differentiate, from first principles, y=x^2

According to first principles, the differential is found as the limit as h->0 of:[f(x+h)-f(x)] / hif we set our f to x^2, then we find that this expression becomes (x^2+2hx+h^2 - x^2)/hWhich simplifies to...
ME
Answered by Milo E. • Maths tutor
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Given that w=x * e^-w use implicit differentiation to show that dw/dx=1/(e^w + x)

Given that w=x e -w use implicit differentiation to show that dw/dx = 1/(e w +x)Answer:Use product rule to simplify:dw/dx = x (de -w /dx) + e -w (dx/dx)Use chain rule to simplify even further:dw/dx = -x e -w...
PG
Answered by Panagiota G. • Maths tutor
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A sequence of numbers have the property that x, 12, y, where x > 0, y > 0, form a geometric sequence while 12, x, 3y form an arithmetic sequence. A)If xy = k, find k. B)Find the value of x and y.

A) Since x, 12, y form a geometric sequence, we know that there exists some real number "r" (common ratio) such that 12=x r and y=12 r. Hence, r= 12/x =y/12 => xy=144 => k=144 (cross multiply...
SS
Answered by Stamatis S. • Maths tutor
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How does proof by induction work?

When using proof by induction we most often prove a statement P for positive integers n. We think about the problem in a domino-toppling fashion. The first step is to write out P(n=1), so inserting 1 for n i...
AC
Answered by Ana C. • Maths tutor
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Prove 2^(n+2) + 3^(2n+1) is a multiple of 7 for all positive integers of n by mathematical induction.

Let P(n) be the proposition that 2 n+2 + 3 2n+1 is a multiple of 7 for all positive integers of n. Let n=12 3 + 3 3 = 8 + 27 = 35 = 7(5)This is divisible by 7. Assume n=k2 k+2 + 3 2k+1 = 7m The above equatio...
EP
Answered by Eashan P. • Maths tutor
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