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There are two ways you can take the derivative of a^x.
Let y = a^x now we're trying to find dy/dx
ln(y) = xln(a) take logs of both sides and use log rules
(dy/dx)*(1/y) = ln(a) take the derivative of both sides using the chain rule on the left hand side.
dy/dx = ln(a)*y multiply both sides by y
dy/dx = ln(a) *a^x realise y= a^x and replace it
Now we're done!
Alternatively we could realise that any exponent can be written as e to the power of something with a log in it.
y = a^x = (e^ln(a))^x just rewritting 'a'
y = e^xln(a) multiplying exponent rule
dy/dx = ln(a)*e^xln(a) take the derivative of both sides using the chain rule for the right hand side
dy/dx = ln(a)*a^x substitute back to get desired result
There are a few different types of integration, the most common being Riemann integration.
Riemann integration allows you to find the area under the graph of a function between two points. Its definition uses something called a limit and it basically says we can approximate the area underneath the graph by adding up areas of rectangles (which is trivially "base times height") so that the width of all the rectangles together go from the first point to the second point and the varying heights of the rectangles goes from the bottom up and "hugs" the curve as best as possible.
Imagine you have a curve and approximate the area underneath it between a and b by finding the area of two rectangles each of width
(b-a)/2, we realise it's quite a poor approximation, but if you make the approximation with 10 rectangles each of width (b-a)/10 and lots of varying heights we realise the approximation is better. Newton and Leibnitz (the independent founders of calculus) then said "what if we take the width of the rectangles to be really really small, so small that the width of the rectangles approaches zero!" then we realise we'd have an infinite number of rectangles to add up all of varying heights and all of width "essentially zero".
What is really going on is that we've said let the width be dx and let us add up all the areas of rectangles as the limit of dx approaches zero. This can be seen in the standard lay out of integrals:
f(x) dx is simply the height of the function multiplied with a width (giving rise to the area of a rectangle) the integral sign at the beginning is an elongated S for "sum", hence you sum up all the areas of rectangles.
Prerequisite for addition formula: https://www.youtube.com/watch?v=iHZmi2MHqWc
Proof of formula for sin(A+B): https://www.youtube.com/watch?v=BEETggUVMrY
Proof of formula for cos(A+B): https://www.youtube.com/watch?v=gTpNnlRxnIY
Proof of formula for tan(A+B): https://www.youtube.com/watch?v=D5VJiL0DMWU
Some C3 Trig stuff: https://www.youtube.com/watch?v=Gcby6WYAVKg
The product rule proof with limits:https://www.youtube.com/watch?v=9FRFRVTUthk