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Use chain rule and implicit differentiation to find dy/dx for y^3 = 1 + 3*x^2, then show that they are equal

When using the Chain Rule, it is best to put our equal in a form where there are no operations performed on the subject. i.e. y = (1 + 3*x 2 ) 1/3 Then to help make it clear what we need to do, state the rul...
JP
Answered by Joshua P. • Maths tutor
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How do I simplify (1 / [1 + cos(x) ] ) + (1 / [1 - cos(x) ] )?

In order to add the fractions together, we must have a common denominator of the fractions. The simplest way to do this is to make the denominators of the equations the product of the original two denomniato...
SC
Answered by Simon C. • Maths tutor
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A curve has the equation: x^4 + 2x -xy - y^3 - 10=0. Find dy/dx in terms of x and y.

We need to differentiate all values with respect to x. Therefore for the first two terms, multiply by the power and then subtract 1 from the original power. Therefore 4(x 4-1 ) + (2)(2x 2-1 ) which gives 4x ...
JG
Answered by John G. • Maths tutor
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A curve has equation y = e^x + 10sin(4x), find the value of the second derivative of this equation at the point x = pi/4.

Firstly, differentiate y with respect to x once to obtain the equation dy/dx = e^x + 40cos(4x). Then differentiate this resultant expression, with respect to x, to acquire a solution for (d^2)y/d(x^2) = e^x ...
JI
Answered by Joe I. • Maths tutor
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A function is defined parametrically as x = 4 sin(3t), y = 2 cos(3t). Find and simplify d^2 y/dx^2 in terms of t and y.

We first need to find dy/dx and we use the fact that dy/dx = dy/dt * dt/dx. So we have dy/dt = -6 sin(3t) and dx/dt = 12 cos(3t). Substituing these in we have dy/dx = -6*sin(3t) 1/(12 cos(3t) which simplifie...
BS
Answered by Barnaby S. • Maths tutor
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