find the derivative of:
a) f(t)=(4t+5)^4
b) g(x)=(x+5)/(x-2)
For a), apply the chain rule:
$\displaystyle \left[f(g(t))\right]^{\prime}=f^{\prime}(g(t))\cdot g^{\prime}(t)$.
In you case, let $\displaystyle f(t)=t^4$ and $\displaystyle g(t)=4t+5$.
For b), apply quotient rule:
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}=\frac{g(x)f^{\prime}(x)-f(x)g^{\prime}(x)}{\left[g(x)\right]^2}$
In your case, let $\displaystyle f(x)=x+5$ and $\displaystyle g(x)=x-2$.
Can you take it from here?
The answer they gave was achieved by using the chain rule. I'm surprised they have the answer in that form, given that you told me that you don't know the rule.
They did it as follows:
$\displaystyle \frac{\,d}{\,dt}\left[\left(4t+5\right)^4\right]=4\left(4t+5\right)^3\cdot\frac{\,d}{\,dt}\left[4t+5\right]=4\left(4t+5\right)^3\cdot 4=16\left(4t+5\right)^3$
Just in case a picture helps...
... where
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to t, and the straight dashed line similarly but with respect to the dashed balloon expression (which is the inner function of the composite and hence subject to the chain rule). So, imagine the dashed balloon is just a variable, like x or u etc...
... to decide the multiplier and the power in the derivative. (In this case, 4 and one less than 4, respectively.)
The second one...
... where the product rule (legs crossed or uncrossed)...
shows the pattern for differentiating a product - in this case, x + 5 times the fraction 1/(x - 2), but we write the fraction as (x - 2) to the power minus one, and apply the chain rule again, to perform the right hand part of the (legs uncrossed) product rule.
Tweak the second balloon on the bottom row...
... so that a suitable common denominator is ready (for simplifying the bottom row) and you can see how the quotient rule is derived from the other two (so you never really need it... and if I've made the first two rules seem really scary, then at least they're all you need!)
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