Given that $$ f(x)=x^{5} $$. Find the value of $$ x $$ if $$ f^{\prime}(x)=405 $$

Question

UpStudy AI · Accepted Answer

We are given $$ f(x) = x^5 $$ and need to find $$ x $$ such that

$$
f'(x) = 405.
$$

**Step 1: Compute the derivative**

Using the power rule, the derivative of $$ f(x) = x^5 $$ is

$$
f'(x) = 5x^4.
$$

**Step 2: Set up the equation**

Set the derivative equal to 405:

$$
5x^4 = 405.
$$

**Step 3: Solve for $$ x^4 $$**

Divide both sides by 5:

$$
x^4 = \frac{405}{5} = 81.
$$

**Step 4: Find $$ x $$**

We need to solve

$$
x^4 = 81.
$$

Taking the fourth root of both sides gives

$$
x = \pm \sqrt[4]{81}.
$$

Since $$ 3^4 = 81 $$, we have

$$
x = \pm 3.
$$

**Final Answer:**

$$
x = 3 \quad \text{or} \quad x = -3.
$$
$$ x = 3 $$ or $$ x = -3 $$.

Given that \( f(x)=x^{5} \). Find the value of \( x \) if \( f^{\prime}(x)=405 \)