Home
Class 12
MATHS
Differentiate a^lnx with respect to x...

Differentiate `a^lnx` with respect to x

Text Solution

AI Generated Solution

The correct Answer is:
To differentiate \( a^{\ln x} \) with respect to \( x \), we can follow these steps: ### Step 1: Define the function Let \( y = a^{\ln x} \). ### Step 2: Use the properties of exponents We can rewrite \( y \) using the exponential function: \[ y = e^{\ln(a^{\ln x})} = e^{\ln x \cdot \ln a} \] ### Step 3: Differentiate using the chain rule Now we differentiate \( y \) with respect to \( x \): \[ \frac{dy}{dx} = \frac{d}{dx}(e^{\ln x \cdot \ln a}) \] Using the chain rule, we get: \[ \frac{dy}{dx} = e^{\ln x \cdot \ln a} \cdot \frac{d}{dx}(\ln x \cdot \ln a) \] ### Step 4: Differentiate the inner function Now we differentiate \( \ln x \cdot \ln a \): \[ \frac{d}{dx}(\ln x \cdot \ln a) = \ln a \cdot \frac{d}{dx}(\ln x) = \ln a \cdot \frac{1}{x} \] ### Step 5: Substitute back into the derivative Substituting this back into our expression for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = e^{\ln x \cdot \ln a} \cdot \left(\ln a \cdot \frac{1}{x}\right) \] ### Step 6: Rewrite in terms of \( y \) Since \( e^{\ln x \cdot \ln a} = a^{\ln x} \), we can write: \[ \frac{dy}{dx} = a^{\ln x} \cdot \frac{\ln a}{x} \] ### Final Answer Thus, the derivative of \( a^{\ln x} \) with respect to \( x \) is: \[ \frac{dy}{dx} = \frac{a^{\ln x} \ln a}{x} \] ---
Promotional Banner

Similar Questions

Explore conceptually related problems

Differentiate logsinx with respect to x :

Differentiate x^tanx with respect to x :

Differentiate tan5x with respect to x :

Differentiate x^sinx with respect to x :

Differentiate e^xsinx with respect to x .

Differentiate e^xlogx with respect to x :

Differentiate tan px with respect to 'x'

differentiate xsinx with respect to x.

Differentiate sinxtanx with respect to x

Differentiate x^(x^x) with respect to x :