Other Differential Functions - AP Calculus AB

Use the quotient rule to find the derivative.

Simplify.

The derivative is .

Use the quotient rule to find the derivative.

Use the power rule to find the derivative.

To find the derivative of this function we must use the Product Rule and the Chain Rule. First we set

and

Now differentiating both of these functions gives

Applying this to the Product Rule gives us,

To differentiate the function properly, we must use the Chain Rule which is,

Therefore the derivative of the function is,

Use the quotient rule to find this derivative.

Recall the quotient rule:

To differentiate this equation we use the Chain Rule.

Using this throughout the equation gives us,

To find the derivative of the function we must use the Chain Rule, which is

Applying this to the function we get,

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

at

x:

y:

The slope is

To find the derivative of the function we must use the Chain Rule

Applying this to the function we are given gives,

To differentiate this problem we will need to use the power rule.

The power rule is, where n is the exponent.

Thus our derivative is,

.

We need to differentiate term by term, applying the power rule,

This gives us

Use the quotient rule to find this derivative.

Remember that the quotient rule is:

Apply this to our problem to get

A line that is normal, that is to say perpendicular to a function at any given point will be normal to this slope of the line tangent to the function at that point.

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Derivative of a natural log:

Trigonometric derivative:

Note that u may represent large functions, and not just individual variables!

Taking the derivative of the function at the point .

The slope of the tangent is

Since the slope of the normal line is perpendicular, it is the negative reciprocal of this value

.

This is a mixture of the product rule and the chain rule:

The first term of the product rule is: cos(x)ln(cos(x))

The second term will have sin(x) but will include the differentiation of the ln(cos(x)), which will require the chain rule:

sin(x) * (1/cos(x)) * (–sin(x)) = –sin2(x)/cos(x)

Combining both we get:

cos(x)ln(cos(x)) – sin2(x)/cos(x)

Now, note that none of the answers are the same as this ;however, we can make an alteration:

sin(x)/cos(x) is the same as tan(x)

Therefore, the answer is: cos(x)ln(cos(x)) – tan(x)sin(x)

Okay, don't be overwhelmed. Take this chain rule one step at a time:

Step 1: Do the sine...

cos(cos(tan(sin(x))))

Step 2: Do the cosine . . .

–sin(tan(sin(x)))

Step 3: do the tangent . . . this is the simple chain rule, so diffentiate the argument as well

sec2(sin(x))cos(x)

Step 4: Multiply them together:

–cos(cos(tan(sin(x)))) * sec2(sin(x))cos(x) * sin(tan(sin(x)))

This is a simple chain rule. The derivative of the secant is secant * tangent; therefore:

f'(x) = sec(x2 + 4x) * tan(x2 + 4x) * (2x + 4)

Distribute everything to get your answer: 2x * sec(x2 + 4x)tan(x2 + 4x) + 4sec(x2 + 4x)tan(x2 + 4x)

Consider this as a chain rule case. Do each step:

Step 1: cos4

4cos3(x2)

Step 2: cos(x2); this can be treated like a normal case of the chain rule

–sin(x2) * 2x

Combining these, we get

–8x * sin(x2)cos3(x2)

f(x) = (100/x2) + (50/x) – 200x2

First, rewrite the equation: 100x–2 + 50x–1 – 200x2

At this point, it is relatively easy to differentiate:

f'(x) = –2 * 100x–3 – 50x–2 – 400x = (–200/x3) – (50/x2) – 400x

Simplify by making x3 the common denominator:

(–200 – 50x – 400x4)/x3

Factor out the common –50 in the numerator to make things look nicer:

–50(4 + x + 8x4)/x3