Differentiation of a Function with Respect to Another Function

Neetesh Kumar | February 08, 2025 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Definition of Derivative with Respect to Another Function
• 2. Formula Used
• 3. Solved Examples
• 4. Conclusion

The derivative of a function with respect to another function is a concept where we differentiate one function in terms of another. This is typically used when both functions are interdependent on a third variable.

Mathematically, if we have two functions $y = f(x)$ and $g(x)$, the derivative of $f(x)$ with respect to $g(x)$, denoted as $\dfrac{dy}{dg}$, is given by:

$\boxed{\frac{dy}{dg} = \frac{\frac{dy}{dx}}{\frac{dg}{dx}}}$

Here:

• $\dfrac{dy}{dx}$ is the derivative of $y$ with respect to $x$.
• $\dfrac{dg}{dx}$ is the derivative of $g$ with respect to $x$.

This is based on the chain rule, and it is commonly used in physics and other fields where variables are functions of other variables.

Example 1: Derivative of a Polynomial with Respect to a Trigonometric Function

Find $\dfrac{d}{d(\sin(x))} \left( x^3 + 2x \right)$.

Solution:

We need to differentiate $x^3 + 2x$ with respect to $\sin(x)$.

Step 1: Find the derivative of $y = x^3 + 2x$ with respect to $x$:

$\dfrac{dy}{dx} = 3x^2 + 2$

Step 2: Find the derivative of $g(x) = \sin(x)$ with respect to $x$:

$\dfrac{dg}{dx} = \cos(x)$

Step 3: Apply the formula:

$\dfrac{dy}{dg} = \dfrac{3x^2 + 2}{\cos(x)}$

Thus, the final answer is: $\boxed{\frac{3x^2 + 2}{\cos(x)}}$.

Example 2: Derivative of an Exponential Function with Respect to a Logarithmic Function

Find $\dfrac{d}{d(\ln(x))} \left( e^{x^2} \right)$.

Solution:

We need to differentiate $e^{x^2}$ with respect to $\ln(x)$.

Step 1: Find the derivative of $y = e^{x^2}$ with respect to $x$:

$\dfrac{dy}{dx} = 2x e^{x^2}$

Step 2: Find the derivative of $g(x) = \ln(x)$ with respect to $x$:

$\dfrac{dg}{dx} = \dfrac{1}{x}$

Step 3: Apply the formula:

$\dfrac{dy}{dg} = \dfrac{2x e^{x^2}}{\frac{1}{x}} = 2x^2 e^{x^2}$

Thus, the final answer is: $\boxed{2x^2 e^{x^2}}$.

Example 3: Derivative of a Trigonometric Function with Respect to an Exponential Function

Find $\dfrac{d}{d(e^{x})} \left( \sin(x) \right)$.

Solution:

We need to differentiate $\sin(x)$ with respect to $e^{x}$.

Step 1: Find the derivative of $y = \sin(x)$ with respect to $x$:

$\dfrac{dy}{dx} = \cos(x)$

Step 2: Find the derivative of $g(x) = e^{x}$ with respect to $x$:

$\dfrac{dg}{dx} = e^{x}$

Step 3: Apply the formula:

$\dfrac{dy}{dg} = \dfrac{\cos(x)}{e^{x}}$

Thus, the final answer is: $\boxed{\frac{\cos(x)}{e^{x}}}$.

Example 4: Derivative of a Rational Function with Respect to a Polynomial Function

Find $\dfrac{d}{d(x^2 + 3)} \left( \dfrac{2x^3}{x^2 + 3} \right)$.

Solution:

We need to differentiate $\dfrac{2x^3}{x^2 + 3}$ with respect to $x^2 + 3$.

Step 1: Find the derivative of $y = \dfrac{2x^3}{x^2 + 3}$ with respect to $x$:

$\dfrac{dy}{dx} = \dfrac{(6x^2)(x^2 + 3) - 2x^3(2x)}{(x^2 + 3)^2}$

Simplifying:

$\dfrac{dy}{dx} = \dfrac{6x^4 + 18x^2 - 4x^4}{(x^2 + 3)^2}$

$\dfrac{dy}{dx} = \dfrac{2x^4 + 18x^2}{(x^2 + 3)^2}$

Step 2: Find the derivative of $g(x) = x^2 + 3$ with respect to $x$:

$\dfrac{dg}{dx} = 2x$

Step 3: Apply the formula:

$\dfrac{dy}{dg} = \dfrac{2x^4 + 18x^2}{(x^2 + 3)^2} \cdot \dfrac{1}{2x}$

Simplifying:

$\dfrac{dy}{dg} = \dfrac{x^3 + 9x}{(x^2 + 3)^2}$

Thus, the final answer is: $\boxed{\frac{x^3 + 9x}{(x^2 + 3)^2}}$.