Hi there
Could someone help me understand the rules for trig and exponential differentiation? I don't know how to use the rules properly so I keep getting questions wrong.
Could someone also explain to me how to draw gradient function graphs as well as drawing the graph given the gradient function?
I would like to make step by step notes on these things
Hey Arisa! Welcome to the forums!
So the rules for trig and exponential differentiation are on your reference sheet, but they all just involve multiplying by the derivative of the inside. So for example, consider \(y=\sin{x^2}\).
The 'inside' bit of the function is \(x^2\), the derivative of which is \(2x\). So we'll be multiplying by that. Now normally, the derivative of \(\sin\) is \(\cos\), so put it all together, and you get \(y'=2x\cos{x^2}\). Notice that the inside bit
doesn't change!
This reflects the rule on your reference sheet, \(\frac{d}{dx}\sin{f(x)}=f'(x)\cos{f(x)}\) - Swap to cos and multiply by the inside derivative!
To improve your working there, why don't you upload some questions you've been struggling with and your attempts at solving them - We can try to show you the way to tackle them
As for sketching gradient functions from regular graphs, again, it is just practice. But here are a few rules to help (possibly pop these in your notes):
- Maximum turning points turn into x-intercepts, going from positive to negative
- Minimum turning points turn into x-intercepts, going from negative to positive
- Inflexions turn into maxima/minima
- Intercepts are useless
I always approach these questions by
marking the sign of the gradient at all points. That is, if it slopes up, mark with a '+'. If down, mark with a '-'. This will show you at a glance where the graph of your gradient function should be above the x-axis and where it should be below.
For going backwards, the rules are reversed a little:
- X-Intercepts turn into maxima/minima
- Turning points turn into inflexions
- Inflexions are useless
Again, the best way to improve here won't be to makes notes (though that is definitely a good thing to do).
You need to practice! Once again, welcome! Be sure to upload any specific questions you have so we can help you out!