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Differentiate with respect to x: i) y=x^3ln(2x) ii) y=(x+sin(2x))^3

i) There first step is to acknowledge the need for both product rule (d(uv)/dx=v.du/dx+u.dv/dx) and chain rule (dz/dx=dz/dy*dy/dx). Here, u=x^3 and v=ln(2x). Therefore, du/dx=3x^2 which is a standard diff...

Answered by Edward M. • Maths tutor

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Differentiate f(x) = (x+3)/(2x-5) using the quotient rule.

For a quotient f(x) = u(x)/v(x), the derivative is f'(x) = (vu'(x) - uv'(x))/v(x)². Applying this to the given function, we find u(x) = x+3 and v(x) = 2x-5. So, u'(x) = 1 and v'(x) = 2...

Answered by Sara R. • Maths tutor

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Integrate 2x/(x^2+3) using the substitution u=x^2+3

u=x² + 3

du/dx=2x

dx=du/2x

2x/(x²+3) dx becomes (2x/u) * (du/2x)

the 2x terms cancel out giving 1/u du

this integrates to ln(u)+c becoming ln(x

Answered by Tom S. • Maths tutor

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Core 1: Given that y = x^4 + x^2+3. Find dy/dx

First what we need to do is we need to think of what the question is asking us to find. In this case it is dy/dx but what is this. This is the rate of change of y with respect to x. For understanding pur...

Answered by David S. • Maths tutor

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How do I do definite integrals?

A definite integral is an indefinite integral with set limits. If we consider integration as a way of calculating the area under a curve, a definite integral calculates the area under the curve between tw...

Answered by Charlie J. • Maths tutor

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