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Further Mathematics

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Differentiate arcsin(2x) using the fact that 2x=sin(y)

Differentiate implicitly on both sides with respect to x to get: 2=cos(y) • (dy/dx). Divide by cos(y) on both sides to get: dy/dx=2/cos(y). Use the trigonometric identity cos^2(y)+sin^2(y)=1 rearranged to...

Answered by Louise O. • Further Mathematics tutor

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f(x)=ln(x). Find the area underneath the curve f(x) between 1 and 2.

We cannot dirrectly intergrate ln(x), so instead we intergrate 1ln(x) using intergration by parts.
The formula for intergration by parts is: ∫ (udv/dx) dx = uv − ∫ vdu/dx dx .
We le...

Answered by Angus T. • Further Mathematics tutor

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Let f(x)=x^x for x>0, then find f'(x) for all x>0.
A common misconception from many students when tackling this problem is that they think the usual 'power rule' works. However, in this case the power is itself a function of x and not just a constant, so ...
Answered by Michael F. • Further Mathematics tutor
1624 Views

Solve the equation 3sinh(2x) = 13 - 3e^(2x), answering in the form 0.5ln(k). where k is an integer
We know that: sinh(2x) = 2sinh(x)cosh(x) , therefore the equation becomes 6sinh(x)cosh(x) = 13 - 3e^2xWe also know that sinh(x) = 0.5*[e^x-e^-x] and that cosh(x) = 0.5*[e
Answered by Matt H. • Further Mathematics tutor
4303 Views
Given that y = arcsinh(x), show that y=ln(x+ sqrt(x^2 + 1) )
In questions involving hyperbolic functions and natural logs, it is often useful to rewrite things in terms of e, since then you might be able to take a natural log at the end of your answer. Here, we can...
Answered by Sam B. • Further Mathematics tutor
4905 Views
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Differentiate arcsin(2x) using the fact that 2x=sin(y)

f(x)=ln(x). Find the area underneath the curve f(x) between 1 and 2.

Let f(x)=x^x for x>0, then find f'(x) for all x>0.

Solve the equation 3sinh(2x) = 13 - 3e^(2x), answering in the form 0.5ln(k). where k is an integer

Given that y = arcsinh(x), show that y=ln(x+ sqrt(x^2 + 1) )

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