Question

find the derivatives of 7^x w r to x^7​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{Let\,\,\,\,u=7^x\,\,\,\,and\,\,\,\,v=x^7}$

Differentiating both the expressions w.r.t. x,

$\tt{\dfrac{du}{dx}=\dfrac{d}{dx}(7^x)\,\,\,\,and\,\,\,\,\dfrac{dv}{dx}=\dfrac{d}{dx}(x^7)}$

$\tt{\implies\dfrac{du}{dx}=7^x\cdot\ln(7)\,\,\,\,and\,\,\,\,\dfrac{dv}{dx}=7\,x^6}$

Now, derivative of u with respect to v will be,

$\sf{\dfrac{du}{dv}=\dfrac{\dfrac{du}{dx}}{\dfrac{dv}{dx}}=\dfrac{7^x\cdot\ln(7)}{7\,x^6}}$

$\sf{\implies\dfrac{du}{dv}=\dfrac{7^{x-1}\cdot\ln(7)}{x^6}}$

$\sf{\implies\dfrac{du}{dv}=7^{x-1}\cdot\,x^{-6}\cdot\ln(7)}$