Question 52 of 70intermediate🔧 ApplyShort Answer1 mark

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,ra, b, c, d, p, q, r and ss are fixed non-zero constants and mm and nn are integers): (ax+b)n(ax + b)^n

Correct Answer

na(ax + b)^{n-1}

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 12 | (Page Page 37)
For More Understanding

Explanation

This question asks for the derivative of (ax+b)n(ax + b)^n. From the context, we know that ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1} for any real number nn. For the composite function (ax+b)n(ax + b)^n, we apply the chain rule along with this power rule. Treating (ax+b)(ax + b) as the inner function, the derivative becomes n(ax+b)n−1n(ax + b)^{n-1} multiplied by the derivative of (ax+b)(ax + b), which is aa. Thus, ddx[(ax+b)n]=na(ax+b)n−1\frac{d}{dx}[(ax + b)^n] = na(ax + b)^{n-1}.

Solution Steps

  1. Step 1: Apply power rule: derivative of unu^{n} is nun−1nu^{n-1}, where u = (ax + b)

  2. Step 2: Multiply by derivative of inner function: d/dx(ax + b) = a

  3. Step 3: Final result: na(ax + b)^{n-1}

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