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Given that z=sin(x)/cos(x), show that dz/dx = sec^2(x).

We have a "fraction" which we wish to differentiate, so we use the quotient rule with u=sin(x) and v=cos(x). This means that d/dx of u/v = (v du/dx - u dv/dx)/(v^2). We have u=sin(x) so du/dx= cos(...
GG
Answered by Gabriela G. • Maths tutor
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Integrate y with respect to x, where y = cos(x)/[1+tan^2(x)]

y = cos(x)/sec 2 (x) = cos 3 (x) y = cos(x)(1-sin 2 (x)) = cos(x) - cos(x) sin 2 (x) -> sin(x) - sin 3 (x)/3 + c
DC
Answered by Daniel C. • Maths tutor
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Using the equation cos(a+b) = cos(a)cos(b) - sin(a)sin(b) or otherwise, show that cos(2x) = 2cos^2(x) - 1.

First let a = b = x such that: cos(a + b) = cos(a)cos(b) - sin(a)sin(b) becomes: cos(x + x) = cos(x)cos(x) - sin(x)sin(x) Leading to: cos(2x) = cos 2 (x) - sin 2 (x) Using the fact that sin 2 (y) + cos 2 (y)...
BH
Answered by Benjamin H. • Maths tutor
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A general function f(x) has the property f(-x)=-f(x). State a trigonometric function with this property and explain using the Maclaurin series expansion for this function why this property holds. Write down the integral in the limits -q to q of f(x) wrt x

Note that this property is the definition of an odd function, or draw a sketch of what this looks like in general about the horizontal axis. E.g. f(x)=sin(x) which has the expansion f(x)=x-((x^3)/3!)+((x^5)/...
JH
Answered by James H. • Maths tutor
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Find the derivative of f(x)=x^3 sin(x)

Find the derivative of f(x)=x 3 sin(x). To do this calculation we need to use the product rule of differentiation: if f(x)=u(x)v(x), then the derivative is f'(x)=u'(x)v(x)+u(x)v'(x). In our case, u(x)=x 3 an...
MM
Answered by Mairi M. • Maths tutor
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