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Explain how Differentiation by the chain rule works

If the expression to be differentiated is a (differentiable) function of another (differentiable) function, then the chain rule must be applied. For example y= f(g(x)), where f and g are both differentiable,...
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Answered by Gwyndaf O. • Maths tutor
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Factorise completely ( x − 4x^3)

(x-4x 3 ) = x(1-4x 2 ) Just taking a factor of x out of both terms Within the inner bracket we have a difference of two squares. 1 2 =1 and (2x) 2 =4x 2 therefore we can write x(1-4x 2 ) = x(1-2x)(1+2x)
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Answered by Edward C. • Maths tutor
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solve the differential equation dy/dx = 6xy^2 given that y = 1 when x = 2

First step is to seperate the variables (EQ1) : (1/y^2) dy = 6x Then we integrate each side seperately giving us (EQ2) : -1/y = 3x^2 + C (remembering to add 1 to the power and divide by the new power) subbin...
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Answered by Dylan M. • Maths tutor
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How would you determine what sort of stationary point this curve has? x^3 - 6x^2 + 9x - 4

I would differentiate it and then turn it into an equation to find the points where the gradient equals zero. With these points at hand, I would take a second derivative, this tells me how the gradient chang...
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Answered by William M. • Maths tutor
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(a) Express 9x+11/(2x+3)(x-1) as partial fractions and (b) find the integral of 9x+11/(2x+3)(x-1) with respect to x

(a) 9x+11/(2x+3)(x-1) = A/2x+3 + B/x-1 9x+11 = (x-1)A + (2x+3)B Setting x=1 gives B=4, setting x=-3/2 gives A=1 so the final answer is 1/(2x+3) + 4/(x-1) (b) 1/2ln|2x+3| + 4ln|x-1| + c
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Answered by Amelia T. • Maths tutor
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