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If a=3a=3, b=7b=7 and c=−5c=-5, verify the following property and identify the name of property. \newlinea×(b+c)=a×b+a×ca \times (b+c) = a \times b + a \times c

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Q. If a=3a=3, b=7b=7 and c=−5c=-5, verify the following property and identify the name of property. \newlinea×(b+c)=a×b+a×ca \times (b+c) = a \times b + a \times c
  1. Differentiate x4x^4: Differentiate each term separately.\newlineddx(x4)=4x3\frac{d}{dx}(x^{4}) = 4x^{3}
  2. Chain rule for ln⁡(2x)\ln(2x): Use the chain rule for ln⁡(2x)\ln(2x), where u=2xu=2x and the derivative of ln⁡(u)\ln(u) is 1u\frac{1}{u}.ddx(ln⁡(2x))=ddu(ln⁡(u))⋅ddx(2x)=12x⋅2=1x\frac{d}{dx}(\ln(2x)) = \frac{d}{du}(\ln(u)) \cdot \frac{d}{dx}(2x) = \frac{1}{2x} \cdot 2 = \frac{1}{x}
  3. Derivative of exe^x: The derivative of exe^{x} is exe^{x}.\newlineddx(ex)=ex\frac{d}{dx}(e^{x}) = e^{x}
  4. Combine derivatives: Combine the derivatives.\newline(d)/(dx)(x4+ln⁡(2x)+ex)=4x3+1x+ex(d)/(dx)(x^{4}+\ln(2x)+e^{x}) = 4x^{3} + \frac{1}{x} + e^{x}

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