Question

Use the limit definition to find the derivative of the function. $$ f(x)=4 x+1 $$

Short answer

The derivative of function \(f(x) = 4x + 1\) is 4, denoted as \(f'(x) = 4\).

Step-by-step solution

Write down the function and the limit definition

The given function is \(f(x) = 4x + 1\). The limit definition of a derivative is \(\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}\). We will substitute the function into this limit definition.

Substitute into the limit definition

Substitute the function into the limit definition: \(\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}\) = \(\lim_{h\to 0} \frac{(4(x+h) + 1)-(4x + 1)}{h}\). This simplifies to \(\lim_{h\to 0} \frac{4h}{h}\).

Simplify the expression

The \(h\) in the numerator and the \(h\) in the denominator cancel out. Now we have \(\lim_{h\to 0}4\). As there is no \(h\) in the function, the limit as \(h\) approaches to 0 is 4. The function is constant.

Write down the derivative

The derivative of function \(f(x)\), denoted \(f'(x)\), is 4, i.e. \(f'(x) = 4\).