# Find the function f given that the slope of the tangent line at any point (x, f(x)) is f '(x) and that the graph of f passes through the given point. f '(x)=5(2x − 1)^{4}, (1, 9)

Efan Halliday 2021-03-05 Answered
Find the function f given that the slope of the tangent line at any point (x, f(x)) is f '(x) and that the graph of f passes through the given point.$f{ }^{\prime }\left(x\right)=5\left(2x-1{\right)}^{4},\left(1,9\right)$
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Derrick

Start by finding the slope of f(x). To do this, integrate f'(x). The integral comes out to $\frac{\left(2x-1{\right)}^{5}}{2}+C$. Now solve for C by solving $f\left(1\right)=\frac{\left(2×1-1{\right)}^{5}}{2}+C=9$, which gives us a C value of $\frac{17}{2}$, thus the function $f\left(x\right)=\frac{\left(2x-1{\right)}^{5}}{2}+\frac{17}{2}$