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here the student needs to use both implicit differentiation and the product rule.

I would differentiate term by term for this problem.

for 2ye^-3x you have to use the proudct rule. uv differ...

We will split this up and differentiate each part separately.

We can differentiate 4x² using our normal rules for differentiating; we multiply the coefficient by the pow...

y=(4x+1)^3*sin(2x) - this is a product of two functions of x. It can be rewritten as y = u(x)*v(x)   ; where u(x) = (4x+1)^3 and v(x) = sin(2x)

Using the product rule: dy/dx = u'(x)*v(x) + v'(x)*u(...

This question must be attempted by integration by parts since it cannot be integrated outright and we can thus change the integral to 1 times lnx. We can then use the formula for integration by parts of I...

First, recall that when integrating, squared trigonometric functions often cause issues. Therefore, use the identity: cos^2(x) = (cos(2x) + 1)/2 to remove this power, giving the integral: ∫ (xcos(2x))/2  ...