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The rate of decay of the mass is modelled by the differential equation dx/dt = -(5/2)x. Given that x = 60 when t = 0, solve the quation for x in terms of t.

(1) Rearrange the equation so that the left hand side is a function of x, and the right hand side is a function of t only.dx/dt = - (5/2) x(1/x)dx = -(5/2)dt(2) Integrate both sidesln(x/A) = -(5/2)t, where A...
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Answered by Joseph C. • Maths tutor
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What are logarithms?

Logarithms allow us to calculate and manipulate indices in relation to regular numbers. The key thing to remember is that " log x y " begs the question "which power of x gives y ". By thi...
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Answered by Alex M. • Maths tutor
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Find the derivative of the equation y = x*ln(x)

y = x*ln(x)Let u = x, v = ln(x) => du/dx = 1, dv/dx = 1/x=> y = uv=> dy/dx = (du/dx)v + u(dv/dx) USING PRODUCT RULETherefore y = ln(x) + 1
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Answered by Owen B. • Maths tutor
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Differentiate z = e^(3y^2+5) with respect to y. (Hint: use chain rule.)

We can find dz/dy using chain rule dz/dy=dz/du x du/dy (1) by defining u=3y^2+5 (since the exponent of e is a function of y we call this function u) and rewrite z=e^u. Then, we find dz/du=e^u (2) and du/dy=6...
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Answered by Sophie H. • Maths tutor
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Given y = 4x/(x^2 +5) find dy/dx, writing your answer as a single fraction in its simplest form

This function is fraction so the easiest method is to use the quotient rule (though the product rule can be used). Recall the quotient rule dy/dx = [vu' - uv']/[v^2]Note, u and v refer to the numerator and d...
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Answered by Laurence C. • Maths tutor
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