Question

Find \(F^{\prime}(x)\). $$ F(x)=\int_{-x}^{x} t^{3} d t $$

Short answer

The derivative \( F^{\prime}(x) \) is \( 0 \).

Step-by-step solution

Understand the Leibniz Rule

According to the Leibniz Rule, the derivative of an integral of a function with variable limits is given by subtracting the function at the lower limit multiplied by the derivative of the lower limit from the function at the upper limit multiplied by the derivative of the upper limit.

Apply the Leibniz Rule

Having defined \( F(x) = \int_{-x}^{x} t^{3} dt \), let's apply the Leibniz Rule. We plug in upper limit \( x \) into \( t^3 \), which is \( x^3 \), and multiply by the derivative of upper limit \( x \), which is \(1\). Then, we plug in lower limit \( -x \) into \( t^3 \), which is \( (-x)^3 = -x^3 \), and multiply by the derivative of the lower limit \( -x \), which is \(-1\). Subtract the second term from the first term.

Simplify the expression

After applying the rule, the obtained result is \( x^3*1 - (-x^3)*(-1) = x^3 - x^3 = 0 \). Therefore, the derivative of the function \( F(x) \) is \( 0 \).