Question

In Exercises 1-4, use the definition \(f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}\) to find the derivative of the given function at the indicated point. $$f(x)=x^{3}+x, a=0$$

Short answer

The derivative of the function \(f(x)=x^{3}+x\) at the point \(a=0\) is 1.

Step-by-step solution

Setup the derivative formula

Start by setting up the limit definition of the derivative: \(f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\). This formula gives the slope of the function at any point.

Substitute \(f(x)\) and \(a\) into the formula

Substitute \(f(x)=x^{3}+x\) and \(a=0\) into the derivative formula to get: \(f'(0) = \lim_{h \to 0} \frac{((0+h)^{3}+(0+h))-(0^{3}+0)}{h}\).

Simplify the expression

The expression simplifies to \(f'(0) = \lim_{h \to 0} \frac{h^{3}+h}{h}\). This can be further simplified by factoring out the \(h\), giving \(f'(0) = \lim_{h \to 0} (h^{2}+1)\).

Evaluate the limit

Now, evaluate the limit by substituting \(h = 0\) into the expression, giving \(f'(0) = 0^{2}+1 = 1\). This is the derivative of the function at the point \(a=0\).