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Differentiate each term using the power rule (d)/(dx)(axⁿ) = anxⁿ−1 and the rule for constants (d)/(dx)(c) = 0.

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2 parts
a

10x

b

1x\frac{1}{x}

1375

Step-by-step solution

12 steps

Q21. Differentiate the following expressions

Part a· 3 steps

  1. Differentiate each term using the power rule ddx(axn)=anxn−1\frac{d}{dx}(ax^n) = anx^{n-1} and the rule for constants ddx(c)=0\frac{d}{dx}(c) = 0.

    ddx(5x2+1)=ddx(5x2)+ddx(1)\frac{d}{dx}(5x^2 + 1) = \frac{d}{dx}(5x^2) + \frac{d}{dx}(1)

  2. Apply the differentiation rules.

    ddx(5x2)=5⋅2x2−1=10x\frac{d}{dx}(5x^2) = 5 \cdot 2x^{2-1} = 10x ddx(1)=0\frac{d}{dx}(1) = 0

  3. Combine the results.

    10x+0=10x10x + 0 = 10x

    The derivative is:

    10x\boxed{10x}

Part b

1x\frac{1}{x}

  1. Use the chain rule for differentiation.

    Let y=ln⁡(u)y = \ln(u) where u=5xu = 5x.
    The chain rule states dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.

  2. Differentiate yy with respect to uu.

    dydu=ddu(ln⁡(u))=1u\frac{dy}{du} = \frac{d}{du}(\ln(u)) = \frac{1}{u}

  3. Differentiate uu with respect to xx.

    dudx=ddx(5x)=5\frac{du}{dx} = \frac{d}{dx}(5x) = 5

  4. Apply the chain rule by substituting u=5xu=5x.

    dydx=1u⋅5=15x⋅5\frac{dy}{dx} = \frac{1}{u} \cdot 5 = \frac{1}{5x} \cdot 5

  5. Simplify the expression.

    55x=1x\frac{5}{5x} = \frac{1}{x}

    The derivative is:

    1x\boxed{\frac{1}{x}}

    Q22. Compute ∑r=150(r+2)\sum_{r=1}^{50} (r+2)

  6. Split the summation into two separate sums.

    ∑r=150(r+2)=∑r=150r+∑r=1502\sum_{r=1}^{50} (r+2) = \sum_{r=1}^{50} r + \sum_{r=1}^{50} 2

  7. Evaluate the first sum, ∑r=150r\sum_{r=1}^{50} r, which is the sum of the first 50 natural numbers.

    Use the formula ∑i=1ni=n(n+1)2\sum_{i=1}^{n} i = \frac{n(n+1)}{2}.

    ∑r=150r=50(50+1)2=50×512\sum_{r=1}^{50} r = \frac{50(50+1)}{2} = \frac{50 \times 51}{2} =25×51=1275= 25 \times 51 = 1275

  8. Evaluate the second sum, ∑r=1502\sum_{r=1}^{50} 2, which is summing the constant 2, 50 times.

    ∑r=1502=50×2=100\sum_{r=1}^{50} 2 = 50 \times 2 = 100

  9. Add the results from Step 2 and Step 3.

    1275+100=13751275 + 100 = 1375

    The computed sum is:

    1375\boxed{1375}
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