5.3.3 Using the Derivative Rules with Transcendental Functions

Question 1

Using the Derivative Rules with Transcendental Functions

Answer 1

• Some functions are combinations of other functions, such as products or quotients. To differentiate these functions, it may be necessary to use several computational techniques, possibly more than once.
• Transcendental functions have unusual derivatives.

Question 2

note

Answer 2

• A transcendental function is a function that cannot be
  expressed in terms of a variable raised to a power. You can use all of the different derivative rules when working with transcendental functions.
• To find the derivative of a composition of a composite function you will need to use the chain rule twice. Each
  additional composite function within a function will require an additional chain rule.
• Notice that the exponent is the outside of this expression. Next is the sine function. The argument of the sine function is the final function.
• Here two functions are combined by multiplication. In addition, the second function is a composite function. To find the derivative you will need the product rule and the chain rule.
• Start with the product rule. Remember that you will need to use the chain rule when asked to find the derivative of the second piece of the product.
• This function is made up of the quotient of two other
  functions. Notice that none of these functions are composite functions.
• Be careful when finding the derivative of the tangent
  function. If you do not remember the formula you can derive it by converting tangent to sines and cosines.

Question 3

Calculate the slope of the line tangent to f (x) = 1 + e^2x at x = 0.

Answer 3

2

Question 4

Find f′(x) if f(x)=cos(2x^2).

Answer 4

f′(x)=−4xsin(2x^2)

Question 5

Suppose f(x)=2^(x^2−2x). Find f′(1).

Answer 5

f′(1)=0

Question 6

Suppose f(x)=log_7x^2.What is the slope of the line tangent to f where x=2?

Answer 6

m=1/ln7

Question 7

Use the definition of the derivative to evaluate the limit limh→0 ln(h+1)/h

Answer 7

lim h→0 ln(h+1)/h=1

Question 8

Let f(x)=1−x^2+3x/2x^4. Rather than use the quotient rule to find f′(x), divide the denominator into the numerator first, and then find f′(x).

Answer 8

f′(x)=2x^2−9x−4/2x^5