Question

Derivative of \(2x^2\) with respect to \(5x^4\) is:

• A. \(\frac{1}{5x}\)
• B. \(-\frac{1}{5x^2}\)
• C. \(\frac{1}{5x^2}\)
• D. \(\frac{1}{5x^3}\)

Answer 1

The Correct Option is C

To find the derivative of \(2x^2\) with respect to \(5x^4\), we use the concept of differentiating a function of another function. Let \(u = 2x^2\) and \(v = 5x^4\). We need to compute \(\frac{du}{dv}\). By the chain rule, \(\frac{du}{dv} = \frac{du}{dx} \times \frac{dx}{dv}\). First, compute \(\frac{du}{dx}\):

\[ \frac{du}{dx} = \frac{d}{dx}(2x^2) = 4x \]

Next, compute \(\frac{dv}{dx}\):

\[ \frac{dv}{dx} = \frac{d}{dx}(5x^4) = 20x^3 \]

Now, find \(\frac{dx}{dv}\) by taking the reciprocal of \(\frac{dv}{dx}\):

\[ \frac{dx}{dv} = \frac{1}{20x^3} \]

Using the chain rule, calculate \(\frac{du}{dv}\):

\[ \frac{du}{dv} = \left(\frac{du}{dx}\right) \times \left(\frac{dx}{dv}\right) = (4x) \times \left(\frac{1}{20x^3}\right) = \frac{4x}{20x^3} = \frac{1}{5x^2} \]

Thus, the derivative of \(2x^2\) with respect to \(5x^4\) is \(\frac{1}{5x^2}\).