Solved: "Use the limit definition to calculate the derivative..."

College
Calculus and Analysis
Differential Calculus
Derivatives

tinfoQ

Answered question

2021-09-21

Use the limit definition to calculate the derivative of the linear function.
k(z)=14z+12

Answer & Explanation

davonliefI

Skilled2021-09-22Added 79 answers

Step 1
Given:
k ( z ) = 14 z + 12.
To find:
The derivative of k ( z ) using limit definition.
Step 2
The limit definition for derivatives:
The derivative of the function f( x ) with respect to x is the function f ’( x ) and is defined as

f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

Step 3
Here, k ( z ) = 14 z + 12.
Therefore,

k^{'} (z) = lim_{h \to 0} \frac{f (z + h) - f (z)}{h}

= lim_{h \to 0} \frac{14 (z + h) + 12 - (14 z + 12)}{h}

= lim_{h \to 0} \frac{14 z + 14 h + 12 - 14 z - 12}{h}

= lim_{h \to 0} \frac{14 h}{h}

= lim_{h \to 0} 14

=14
Therefore, The derivative of k ( z) using limit definition is k ’( z ) = 14.

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