Differentiate (2x+2)(x-5)^8 with respect to x. 8(2x+2)(x-5)^7+2(x-5)^8 (2x+2)(x-5)^7+2(x-5)^8 2(x-5)^8 16(x-5)^7

To differentiate the given expression $(2x+2)(x-5)^{8}$ with respect to $x$, we can use the product rule of differentiation.<br /><br />The product rule states that if we have two functions $u(x)$ and $v(x)$, then the derivative of their product is given by:<br /><br />$\frac{d}{dx}(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)$<br /><br />In this case, let $u(x) = 2x+2$ and $v(x) = (x-5)^{8}$.<br /><br />Now, let's differentiate each function separately:<br /><br />$u'(x) = 2$<br /><br />$v'(x) = 8(x-5)^{7}$<br /><br />Now, applying the product rule:<br /><br />$\frac{d}{dx}((2x+2)(x-5)^{8}) = u'(x)v(x) + u(x)v'(x)$<br /><br />Substituting the values of $u(x)$, $u'(x)$, $v(x)$, and $v'(x)$:<br /><br />$\frac{d}{dx}((2x+2)(x-5)^{8}) = 2(x-5)^{8} + (2x+2) \cdot 8(x-5)^{7}$<br /><br />Simplifying the expression:<br /><br />$\frac{d}{dx}((2x+2)(x-5)^{8}) = 2(x-5)^{8} + 16(x-5)^{7}$<br /><br />Therefore, the correct answer is:<br /><br />$\frac{d}{dx}((2x+2)(x-5)^{8}) = 2(x-5)^{8} + 16(x-5)^{7}$