All Calculus 1 Resources
Example Questions
Example Question #247 : Functions
Differentiate the function:
We evaluate this derivative using the quotient rule:
,
.
Apply the above formula:
, which is our final answer.
Example Question #61 : How To Find Differential Functions
What is the slope of the line tangent to f(x) = x4 – 3x–4 – 45 at x = 5?
400.096
355.00384
500.00384
422.125
355.096
500.00384
First we must find the first derivative of f(x).
f'(x) = 4x3 + 12x–5
To find the slope of the tangent line of f(x) at 5, we merely have to evaluate f'(x) at 5:
f'(5) = 4*53 + 12* 5–5 = 500 + 12/3125 = 500.00384
Example Question #249 : Functions
Solve for when
using the identity:
Example Question #251 : Functions
Differentiate
The Quotient Rule applies when differentiating quotients of functions. Here, equals the quotient of two functions, and . Let and . (Think: is the "low" function or denominator and is the "high" function or numerator.) The Quotient Rule tells us to multiply the "low" function by the derivative of the "high" function, subtract the product of the "high" function and the derivative of the "low" function, and then divide the result by the square of the "low" function. In other words,
Here, so . Similarly, so .
Then
Factoring out from the numerator gives
inverts the order of the numerator, subtracting from .
adds the products in the numerator, rather than subtracting them.
fails to square the denominator.
Example Question #64 : How To Find Differential Functions
Differentiate
The Quotient Rule applies when differentiating quotients of functions. Here, equals the quotient of two functions, and . Let and . (Think: is the "low" function or denominator and is the "high" function or numerator.) The Quotient Rule tells us to multiply the "low" function by the derivative of the "high" function, subtract the product of the "high" function and the derivative of the "low" function, and then divide the result by the square of the "low" function. In other words,
Here, so . Similarly, so .
Then
Factoring out gives
inverts the order of the numerator, subtracting from .
adds the products in the numerator, rather than subtracting them.
fails to square the denominator.
Example Question #251 : Differential Functions
Differentiate
The Quotient Rule applies when differentiating quotients of functions. Here, equals the quotient of two functions, and . Let and . (Think: is the "low" function or denominator and is the "high" function or numerator.) The Quotient Rule tells us to multiply the "low" function by the derivative of the "high" function, subtract the product of the "high" function and the derivative of the "low" function, and then divide the result by the square of the "low" function. In other words,
Here, so . Similarly, so .
Then
inverts the order of the numerator, subtracting from .
adds the products in the numerator, rather than subtracting them.
fails to square the denominator.
Example Question #66 : How To Find Differential Functions
Differentiate
The Chain Rule applies when differentiating compositions of functions. Here, equals the composition of two functions, and . Let and . Then and the Chain Rule tells us to differentiate the outside function and multiply the result by the derivative of the inside function . In other words, . Note that the inside function is left untouched when the outside function is differentiated. Here, and . Remember, roots can (and should) be rewritten as fractional exponents, so becomes which is then differentiated like any other exponent. So
is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.
is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.
is a misapplication of the Power Rule which fails to subtract 1 from the original exponent.
Example Question #67 : How To Find Differential Functions
Differentiate
The Chain Rule applies when differentiating compositions of functions. Here, equals the composition of two functions, and . Let and . Then and the Chain Rule tells us to differentiate the outside function and multiply the result by the derivative of the inside function . In other words, . Note that the inside function is left untouched when the outside function is differentiated. Here, and , so which simplifies to .
is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.
is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.
is a misapplication of the Chain Rule which substitutes the derivative of the inside function for the original inside function rather than multiplying the derivative of the outside function by the derivative of the inside function.
Example Question #68 : How To Find Differential Functions
Differentiate
The Chain Rule applies when differentiating compositions of functions. Here, equals the composition of two functions, and . Let and . Then and the Chain Rule tells us to differentiate the outside function and multiply the result by the derivative of the inside function . In other words, . Note that the inside function is left untouched when the outside function is differentiated. Here, and , so which simplifies to .
is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.
is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.
is a misapplication of the Chain Rule which fails to preserve the original inside function when differentiating the outside function.
Example Question #69 : How To Find Differential Functions
Differentiate
The Chain Rule applies when differentiating compositions of functions. Here, equals the composition of two functions, and . Let and . Then and the Chain Rule tells us to differentiate the outside function and multiply the result by the derivative of the inside function . In other words, . Note that the inside function is left untouched when the outside function is differentiated. Here, and , so which simplifies to .
is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.
is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.
is a misapplication of the Chain Rule which adds the derivative of the outside function to an incorrect derivation of the inside function.
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