The problem is: tansqrt(5t). I get that I need to use the chain rule but I think I'm messing it up.
So I did this: tansqrt(5t) = sec^2sqrt(5t) * 1/2(tt)^-1/2 = sec^2sqrt(5t)/2sqrt(5t).
However, this is the wrong answer. What am I missing?
The problem is: tansqrt(5t). I get that I need to use the chain rule but I think I'm messing it up.
So I did this: tansqrt(5t) = sec^2sqrt(5t) * 1/2(tt)^-1/2 = sec^2sqrt(5t)/2sqrt(5t).
However, this is the wrong answer. What am I missing?
I always use Leibnitz notation when using the Chain Rule. You can have as many links in your chain as you need. Here we can see the 5t function inside a square root function inside a tangent function. Since it's three functions deep, we need three links in the chain.
$\displaystyle \displaystyle \begin{align*} u &= 5t \implies y = \tan{ \left( \sqrt{u} \right) } \\ \\ v &= \sqrt{u} \implies y = \tan{(v)} \\ \\ \frac{du}{dx} &= 5 \\ \\ \frac{dv}{du} &= -\frac{1}{2\sqrt{u}} = -\frac{1}{2\sqrt{ 5t }} \\ \\ \frac{dy}{dv} &= \sec^2{(v)} = \sec^2{ \left( \sqrt{u} \right) } = \sec^2{ \left( \sqrt{ 5t } \right) } \\ \\ \frac{dy}{dx} &= \frac{du}{dx} \cdot \frac{dv}{du} \cdot \frac{dy}{dv} \\ &= 5 \left( -\frac{1}{2\sqrt{ 5t} } \right) \sec^2{ \left( \sqrt{5t} \right) } \\ &= -\frac{5\sec^2{ \left( \sqrt{5t} \right) }}{2\sqrt{5t}} \end{align*}$