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The key fact here is to remember the relationship between the gradients of lines which are parallel and perpendicular. Two lines that are parallel will always have the same gradient. So if we have a line ...

Initially this looks unlike all the other differentiation questions and seems unsolvable. However the expression 2^x can be rewritten in an equivalent form that will allow us to use the differentiation ru...

Integration by parts can be considered as the inverse method of differentiation using the product rule. With the product rule we have: d(fg)/dx = f(dg/dx) + g*(df/dx) where f and g are functions ...

To differentiate a function of x and y, you must differentiate x as you would ordinarily, and then differentiate y as you would normally, but multiply the differentiated term by dy/dx. For terms with x an...

We use the chain rule. Let u(x)=exp(x), v(x)=x^1/2, w(x) = tan(x).
Then f(x) = u(v(w(x))). So by the chain rule, f'(x) = u'(v(w(x)))*(v(w(x)))'.
u'(x) = exp(x).
By the chain rule...