$$ f(x)=\int_{4}^{x} \frac{5}{t}+\cos (t) d t $$ What is $$ \frac{d f}{d x} ? $$

Question

UpStudy AI · Accepted Answer

To find the derivative of the function

$$
f(x) = \int_{4}^{x} \left( \frac{5}{t} + \cos(t) \right) dt,
$$

we can apply the **Fundamental Theorem of Calculus**, which states that if

$$
F(x) = \int_{a}^{x} g(t) \, dt,
$$

then the derivative $$ F'(x) $$ is simply $$ g(x) $$.

In this case, the integrand is

$$
g(t) = \frac{5}{t} + \cos(t).
$$

Therefore, the derivative of $$ f(x) $$ with respect to $$ x $$ is

$$
\frac{df}{dx} = \frac{5}{x} + \cos(x).
$$

**Final Answer:**

$$
\frac{df}{dx} = \frac{5}{x} + \cos(x)
$$
$$ \frac{df}{dx} = \frac{5}{x} + \cos(x) $$

\( f(x)=\int_{4}^{x} \frac{5}{t}+\cos (t) d t \) What is \( \frac{d f}{d x} ? \)