Find the Derivative - d/dx fifth root of 8x^3+9x

Problem

d()/d(x)√(5,8*x3+9*x)

Solution

1. Rewrite the radical expression using a fractional exponent to make it easier to apply the power rule.

√(5,8*x3+9*x)=(8*x3+9*x)(1/5)

2. Apply the chain rule, which states that the derivative of ƒ*(g(x)) is ƒ′*(g(x))⋅g(x)′

d()/d(x)*(8*x3+9*x)(1/5)=1/5*(8*x3+9*x)(1/5−1)⋅d()/d(x)*(8*x3+9*x)

3. Differentiate the inner function 8*x3+9*x using the power rule.

d(8*x3+9*x)/d(x)=24*x2+9

4. Substitute the inner derivative back into the chain rule expression and simplify the exponent.

1/5*(8*x3+9*x)(−4/5)⋅(24*x2+9)

5. Simplify the expression by moving the term with the negative exponent to the denominator and rewriting it in radical form.

(24*x2+9)/(5*(8*x3+9*x)(4/5))

Final Answer

d(√(5,8*x3+9*x))/d(x)=(24*x2+9)/(5√(5,(8*x3+9*x)4))

Want more problems? Check here!