Find the Derivative - d/dx y = square root of 1+8x

Problem

d()/d(x)√(,1+8*x)

Solution

1. Rewrite the square root as a power to prepare for differentiation.

√(,1+8*x)=(1+8*x)(1/2)

2. Apply the power rule and the chain rule, where the outer function is u(1/2) and the inner function is u=1+8*x

d(1+8*x)/d(x)=1/2*(1+8*x)(−1/2)⋅d(1+8*x)/d(x)

3. Differentiate the inner function 1+8*x with respect to x

d(1+8*x)/d(x)=8

4. Multiply the results together.

1/2*(1+8*x)(−1/2)⋅8=4*(1+8*x)(−1/2)

5. Simplify the expression by moving the negative exponent to the denominator and converting back to radical form.

4*(1+8*x)(−1/2)=4/√(,1+8*x)

Final Answer

d(√(,1+8*x))/d(x)=4/√(,1+8*x)

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