Answer:
[tex]f'(x)=\dfrac{5}{2\sqrt{x}}[/tex]
Step-by-step explanation:
To find the derivative of f(x) = 5√x, start by expressing the square root as an exponent:
[tex]f(x)=5x^{\frac{1}{2}}[/tex]
Now, apply the power rule:
[tex]\boxed{\begin{array}{c}\underline{\textsf{Power Rule for Differentiation}}\\\\\textsf{If}\;f(x)=x^n\;\;\textsf{then}\;\;f'(x)=nx^{n-1}\\\end{array}}[/tex]
Therefore:
[tex]f'(x)=\dfrac{1}{2} \cdot 5x^{\left(\frac{1}{2}-1\right)}[/tex]
Simplify:
[tex]f'(x)=\dfrac{5}{2}x^{-\frac{1}{2}}[/tex]
Express the derivative in terms of the square root:
[tex]f'(x)=\dfrac{5}{2\sqrt{x}}[/tex]
Therefore, the derivative of f(x) = 5√x is:
[tex]\Large\boxed{\boxed{f'(x)=\dfrac{5}{2\sqrt{x}}}}[/tex]
To find the derivative of f(x) = 5sqrt(x), we express the square root as x^(1/2), apply the power rule, and then multiply by the constant 5 to get f'(x) = (5/2)/sqrt(x).
The derivative of f(x) = 5sqrt(x) using the shortcut rules of derivatives can be calculated by expressing the square root as a fractional exponent. The square root of x, which is sqrt(x), can be rewritten as x^(1/2). Using the power rule, the derivative of x^(1/2) is (1/2)x^(1/2 - 1), which simplifies to (1/2)x^(-1/2). Multiplying this by the constant 5 gives us f'(x) = (5/2)*x^(-1/2), which can be expressed as (5/2)sqrt(x). Therefore, the final answer is f'(x) = (5/2)/sqrt(x).
