## Review of basic derivative rules

In calculus, the derivative is a way to find the rate of change of a function with respect to one of its variables. In other words, the derivative tells us how a function is changing. The derivative can be thought of as a slope of a function, and just like with any slope, the steeper the slope, the faster the function is changing.

### Product rule

To find the equation of the tangent line to fxx3 x2 x 1 at x4, we need to use the product rule. This states that if f and g are differentiable functions, then their product is also a differentiable function, and its derivative is given by:

(f * g)’ = f’ * g + f * g’

Using this rule, we can calculate the derivative of our original function as follows:

f'(x) = 3 * 2 * x + 3 * x^2 + 1 * 1 = 6x + 9x^2 + 1

Now that we have the derivative of our function, we can plug in our value for x (4) to find the equation of the tangent line:

f'(4) = 6(4) + 9(4)^2 + 1 = 24 + 144 + 1 = 169

Therefore, the equation of the tangent line to fxx3 x2 x 1 at x4 is y=169*(x-4)+f(4).

### Quotient rule

The quotient rule is a rule for finding the derivative of a function that is the quotient of two other functions. The rule states that the derivative of the function f(x)/g(x) is equal to (f'(x)g(x)-f(x)g'(x))/(g(x))^2. So, using this rule, the derivative of fxx3 x2 x 1 at x4 would be ((3*4^2)*1-(4^3)*1)/(1)^2, or 36.

### Power rule

One of the most basic derivative rules is the power rule. This rule states that, for any differentiable function f(x) and any real number n, we have:

f'(x) = nf(x)

This means that, to find the derivative of f(x), we just need to multiply the function by n. For example, if we have f(x) = x2, then we can find its derivative using the power rule:

f'(x) = 2×2

We can also use the power rule to find the derivative of functions that are not polynomials. For example, let’s find the derivative of f(x) = 3×2 – 2x + 5:

f'(x) = 6x – 2

### Chain rule

In calculus, the chain rule is a formula used to compute the derivative of a composite function. If the composite function is differentiable, then the chain rule gives a formula for the derivative in terms of the derivatives of its component functions.

The chain rule is one of the most frequently used rules in calculus; it allows one to differentiate functions that are built up from other functions using multiplication, composition, and inversions. Many common functions are defined using these operations, so the chain rule allows one to compute their derivatives. It also has applications outside of calculus, including in probability theory and statistics.

The simplest form of the chain rule states that if f is a differentiable function of g and g is a differentiable function of x, then f is a differentiable function of x and

`(f ∘ g)'(x) = f'(g(x))g'(x).`

Here ∘ denotes function composition.

## Implicit differentiation

In implicit differentiation, we differentiate a function implicitly with respect to x. This means that we treat y as a function of x and differentiate as if we were dealing with a single variable function. We can use implicit differentiation to find the equation of the tangent line to a curve at a given point.

### What is implicit differentiation?

In mathematics, implicit differentiation is a method of finding the derivative of a function that is not explicitly given. This means that the function is not defined in terms of x, but rather in terms of some other variable (or variables). For example, the equation y2=x3 defines a function of x, but it is not explicitly given in terms of x. To find the derivative of this function with respect to x, we would need to use implicit differentiation.

### How to implicit differentiate

Differentiating implicitly means that we differentiate without solving for y first. This can be useful when we want to find the equation of the tangent line to a curve without having to find the y coordinate first. For example, consider the following curve:

f(x) = x^3 + x^2 + x

To find the equation of the tangent line to this curve at x = 4, we would need to take the derivative of f(x) with respect to x and then evaluate it at x = 4. However, we can also implicitly differentiate f(x) with respect to x and then solve for y’ at x = 4. This gives us the following:

f'(x) = 3x^2 + 2x + 1

y’ = f'(4) = 3(4)^2 + 2(4) + 1 = 57

Now that we have y’, we can use it to write the equation of the tangent line as follows:

y – f(4) = 57(x – 4)

y – (4^3 + 4^2 + 4) = 57(x – 4) —> y = 57x – 243

## Tangent lines

The equation of the tangent line to a graph at a certain point is the equation of a line that just barely touches the graph at that point. A tangent line is perpendicular to the radius of the curve at the point of tangency. The slope of the tangent line at a point on a graph is the slope of the graph at that point. The equation of the tangent line to fxx3 x2 x 1 at x4 is y=12x-11.

### What is a tangent line?

In mathematics, a tangent line (or simply tangent) to a curve y = f(x) at a point x0 on the curve is a line that just touches the curve at that point. More precisely, a tangent line is a line that passes through the point x0 on the curve y = f(x) and has slope f'(x0).

### How to find the equation of a tangent line

There are a few different ways to find the equation of a tangent line, but one of the most common is using the point-slope form. To use this method, you’ll need to know the coordinates of the point where the tangent line intersects the graph of the function (x,y), as well as the slope of the tangent line.

The point-slope form of a line is:

y – y1 = m(x – x1)

where m is the slope of the line and (x1,y1) is the coordinates of the point of intersection.

For our example, we would plug in our values as follows:

y – 3 = m(x – 4)

Now we just need to solve for m, which we can do by plugging in a known point on the graph and solving for m. Let’s say we know that (2,9) is also on the tangent line. We can plug this in and solve for m:

9 – 3 = m(2 – 4)

## worked example

Take the derivative of f(x) = 3x^2 + 2x + 1 using the power rule.

### Find the equation of the tangent line to fxx3 x2 x 1 at x4

To find the equation of the tangent line to fxx3 x2 x 1 at x4, we need to find the first derivative of the function. This can be done using the following steps:

- Take the derivative of each term in the function.
- Simplify the resulting expression.

The first derivative of fxx3 x2 x 1 is 3×2(2x-1)+x(3x-1).

Plugging in 4 for x, we get 3(4)2(2(4)-1)+4(3(4)-1), which simplifies to 48+48, or 96.

Therefore, the equation of the tangent line is y=96x-384.