Question 50 of 70intermediate🔧 ApplyShort Answer1 mark

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ax4−bx2+cos⁡x\frac{a}{x^4} - \frac{b}{x^2} + \cos x

Correct Answer

−4ax5+2bx3−sin⁡x-\frac{4a}{x^5} + \frac{2b}{x^3} - \sin x

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 10 | (Page Page 37)
For More Understanding

Explanation

This question tests the application of the power rule for differentiation on algebraic terms and the standard derivative of trigonometric functions. The expression contains three terms that can be differentiated independently using the sum/difference rule.

Solution Steps

  1. Step 1: Rewrite the expression using negative exponents: ax4−bx2+cos⁡x=ax−4−bx−2+cos⁡x\frac{a}{x^4} - \frac{b}{x^2} + \cos x = ax^{-4} - bx^{-2} + \cos x

  2. Step 2: Apply the power rule ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1} to each algebraic term: ddx(ax−4)=a⋅(−4)x−5=−4ax5\frac{d}{dx}(ax^{-4}) = a \cdot (-4)x^{-5} = -\frac{4a}{x^5} and ddx(−bx−2)=−b⋅(−2)x−3=2bx3\frac{d}{dx}(-bx^{-2}) = -b \cdot (-2)x^{-3} = \frac{2b}{x^3}

  3. Step 3: Apply the standard derivative of cosine: ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x) = -\sin x

  4. Step 4: Combine all terms: −4ax5+2bx3−sin⁡x-\frac{4a}{x^5} + \frac{2b}{x^3} - \sin x

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