# What is the equation of the line tangent to the curve y=(x^2-35)^7 at x=6?

May 25, 2015

The tangent line in a point to a function, knowing its ascissa is:

$y - f \left({x}_{0}\right) = f ' \left({x}_{0}\right) \left(x - {x}_{0}\right)$.

So:

$f \left({x}_{0}\right) = f \left(6\right) = {\left(36 - 35\right)}^{7} = 1$

and

$f ' \left(x\right) = 7 {\left({x}^{2} - 36\right)}^{6} \cdot 2 x \Rightarrow$

$f ' \left(6\right) = 7 {\left(36 - 35\right)}^{6} \cdot 2 \cdot 6 = 7 \cdot 1 \cdot 12 = 84$.

The tangent line is:

$y - 1 = 84 \left(x - 6\right) \Rightarrow y = 84 x - 503$.

May 25, 2015

Try this: