# Derivatives Help: Find h'(t) & h''(t) of Let h(t)=tan(3t+7)

In summary, the conversation involves finding the derivatives of the function h(t)=tan(3t+7). The first derivative, h'(t), is found to be 3(sec(3t+7))^2. The conversation then continues to discuss how to find the second derivative, h''(t), using the chain rule and the formula for differentiating secant functions.

## Homework Statement

Let h(t) = tan(3t+7)

Find h'(t) and h''(t)

I found h'(t) which is equal to 3(sec(3t+7))^2

But I can't seem to find h''(t)

How do I find the derivative of this? Could someone please teach me?

Is it a composition function? If it is, I think I see 4 functions.

There are at least two ways to do it. The key observation is that sec(x)=1/cos(x). That should give you the answer using standard tools for handling derivatives of functions.

Ibix said:
There are at least two ways to do it. The key observation is that sec(x)=1/cos(x). That should give you the answer using standard tools for handling derivatives of functions.

Yeah, I knew that sec(x) = 1/cos(x)

and I still couln't figure out what to do

Maybe if you substitute 1/cos for sec in h'(t) that will give you a clue.

Or you can differentiate sec(u) directly, using the formula d/dt(sec(u)) = sec(u)tan(u) * du/dt.

You have already used what I would call the Chain Rule:
$$\frac{d}{dx}f_1(f_2(x))=f_1'(f_2(x))f_2'(x)$$
to get h'(x). All you need to realize is that you can nest functions as deep as you like - just replace $x$ with $f_3(x)$ throughout and tack $f_3'(x)$ on the end:
$$\frac{d}{dx}f_1(f_2(f_3(x)))=f_1'(f_2(f_3(x)))f_2'(f_3(x))f_3(x)$$
Then you need to work out what each of the $f$s is here and dive in.

## 1. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function with respect to its input variable. In simpler terms, it measures how much a function's output changes when its input variable is changed.

## 2. How do you find the derivative of a function?

To find the derivative of a function, we use a mathematical process called differentiation. This involves applying specific rules and formulas to the function to determine its derivative. In this case, we would use the chain rule to find h'(t) and h''(t) for the given function.

## 3. What does h'(t) represent?

h'(t) represents the first derivative of the function h(t). It tells us the instantaneous rate of change of h(t) at any given point on the graph.

## 4. How do you find h'(t) for a given function?

To find h'(t), we use the chain rule, which states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. In this case, we would apply the chain rule to the function h(t)=tan(3t+7) to find h'(t).

## 5. How do you find h''(t) for a given function?

To find h''(t), we first find h'(t) using the chain rule, and then we use the same process again to find the derivative of h'(t). This is known as taking the second derivative of the function. In this case, we would apply the chain rule twice to the function h(t)=tan(3t+7) to find h''(t).