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Integrate by parts:u = ln(x) u' = 1/x v' = 1 v = xBy parts formula: uv - ∫u'v dx Therefore we have: xln(x) - ∫x1/x dx = xln(x) - ∫1 dx = xln(x) - x (+c)

The chain rule is used to find the derivative of an expression in the form h(f(x)) where you have a function in terms of a function of x for example:
h(f(x)) = 2(3x+1)^3 where f(x) = 3x+1
In or...

sin(2x)=sin(x+x) =sin(x)cos(x)+cos(x)sin(x) =2*sin(x)*cos(x); cos(2x)=cos(x+x) =cos(x)cos(x)-sin(x)sin(x) =[cos(x)]²-[sin(x)]² Now let x+y=a and x-y=b, then x=(a+b)/2 and y=(a-b)/2. ...

Use implicit differentiation on original equation-
12x + 3x(dy/dx) + 3y - 2y(dy/dx) = 0
dy/dx= -12x -3y/(3x-2y) at A, x= -5 and y= 10 therefore, dy/dx=-6/7

To find the normal of t...

This requires the chain rule and the product rule to be used to differentiate the function. The substitution u = x² can be used to make this easier. Using this, du/dx = 2x and y = (u-5)e^u<...