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So I want to obtain the following form:

$$\nabla f(x+p) = \nabla f(x) + \int^1_0 \nabla^2 f(x+tp)p \,dt$$ for $f$ twice continuously differentiable and $t \in (0,1)$.

Taylor's theorem in integral form is:

$$f(x) = f(a) + \int^x_a \nabla f(t) (x-t)\,dt$$ from here: https://www.math.washington.edu/~folland/Math425/taylor2.pdf (p.3)

My approach is:

$$f(x+p) = f(x) + \int^{x+p}_x \nabla f(t) (x+p-t)\,dt$$ Reformulated as: $$f(x+p) = f(x) + \int^1_0 \nabla f(x+tp)(1-t)p\,dt$$ $$= f(x) + \int^1_0 \nabla f(x+tp) p \,dt - \int^1_0 \nabla f(x+tp) t p \,dt$$

But I am not sure how to take the derivatives of the integrals with respect to x; am I at a dead end? Should I instead, just begin with $\nabla f(x)$ in place of $f(x)$ when I apply Taylor's theorem?

So I want to obtain the following form: For $x, p \in R^n$

$$\nabla f(x+p) = \nabla f(x) + \int^1_0 \nabla^2 f(x+tp)p \,dt$$ for $f$ twice continuously differentiable and $t \in (0,1)$.

Taylor's theorem in integral form is:

$$f(x) = f(a) + \int^x_a \nabla f(t) (x-t)\,dt$$ from here: https://www.math.upenn.edu/~kazdan/361F15/Notes/Taylor-integral.pdf

My approach is:

$$f(x+p) = f(x) + \int^{x+p}_x \nabla f(t) (x+p-t)\,dt$$ Reformulated as: $$f(x+p) = f(x) + \int^1_0 \nabla f(x+tp)(1-t)p\,dt$$ $$= f(x) + \int^1_0 \nabla f(x+tp) p \,dt - \int^1_0 \nabla f(x+tp) t p \,dt$$

But I am not sure how to take the derivatives of the integrals with respect to x; am I at a dead end? Should I instead, just begin with $\nabla f(x)$ in place of $f(x)$ when I apply Taylor's theorem?

So I want to obtain the following form:

$$\nabla f(x+p) = \nabla f(x) + \int^1_0 \nabla^2 f(x+tp)p \,dt$$ for $f$ twice continuously differentiable and $t \in (0,1)$.

Taylor's theorem in integral form is:

$$f(x) = f(a) + \int^x_a \nabla f(t) (x-t)\,dt$$ from here: http://math.fau.edu/schonbek/DiffGeo/dgsp11_notes_2.pdf (p2)

My approach is:

$$f(x+p) = f(x) + \int^{x+p}_x \nabla f(t) (x+p-t)\,dt$$ Reformulated as: $$f(x+p) = f(x) + \int^1_0 \nabla f(x+tp)(1-t)p\,dt$$ $$= f(x) + \int^1_0 \nabla f(x+tp) p \,dt - \int^1_0 \nabla f(x+tp) t p \,dt$$

But I am not sure how to take the derivatives of the integrals with respect to x; am I at a dead end? Should I instead, just begin with $\nabla f(x)$ in place of $f(x)$ when I apply Taylor's theorem?

So I want to obtain the following form:

$$\nabla f(x+p) = \nabla f(x) + \int^1_0 \nabla^2 f(x+tp)p \,dt$$ for $f$ twice continuously differentiable and $t \in (0,1)$.

Taylor's theorem in integral form is:

$$f(x) = f(a) + \int^x_a \nabla f(t) (x-t)\,dt$$ from here: https://www.math.washington.edu/~folland/Math425/taylor2.pdf (p.3)

My approach is:

$$f(x+p) = f(x) + \int^{x+p}_x \nabla f(t) (x+p-t)\,dt$$ Reformulated as: $$f(x+p) = f(x) + \int^1_0 \nabla f(x+tp)(1-t)p\,dt$$ $$= f(x) + \int^1_0 \nabla f(x+tp) p \,dt - \int^1_0 \nabla f(x+tp) t p \,dt$$

But I am not sure how to take the derivatives of the integrals with respect to x; am I at a dead end? Should I instead, just begin with $\nabla f(x)$ in place of $f(x)$ when I apply Taylor's theorem?

So I want to obtain the following form:

$$\nabla f(x+p) = \nabla f(x) + \int^1_0 \nabla^2 f(x+tp)p \,dt$$ for $f$ twice continuously differentiable and $t \in (0,1)$.

Taylor's theorem in integral form is:

$$f(x) = f(a) + \int^x_a \nabla f(t) (x-t)\,dt$$ from here: https://www.math.upenn.edu/~kazdan/361F15/Notes/Taylor-integral.pdf

My approach is:

$$f(x+p) = f(x) + \int^{x+p}_x \nabla f(t) (x+p-t)\,dt$$ Reformulated as: $$f(x+p) = f(x) + \int^1_0 \nabla f(x+tp)(1-t)p\,dt$$ $$= f(x) + \int^1_0 \nabla f(x+tp) p \,dt - \int^1_0 \nabla f(x+tp) t p \,dt$$

But I am not sure how to take the derivatives of the integrals with respect to x; am I at a dead end? Should I instead, just begin with $\nabla f(x)$ in place of $f(x)$ when I apply Taylor's theorem?

So I want to obtain the following form:

$$\nabla f(x+p) = \nabla f(x) + \int^1_0 \nabla^2 f(x+tp)p \,dt$$ for $f$ twice continuously differentiable and $t \in (0,1)$.

Taylor's theorem in integral form is:

$$f(x) = f(a) + \int^x_a \nabla f(t) (x-t)\,dt$$ from here: http://math.fau.edu/schonbek/DiffGeo/dgsp11_notes_2.pdf (p2)

My approach is:

$$f(x+p) = f(x) + \int^{x+p}_x \nabla f(t) (x+p-t)\,dt$$ Reformulated as: $$f(x+p) = f(x) + \int^1_0 \nabla f(x+tp)(1-t)p\,dt$$ $$= f(x) + \int^1_0 \nabla f(x+tp) p \,dt - \int^1_0 \nabla f(x+tp) t p \,dt$$

But I am not sure how to take the derivatives of the integrals with respect to x; am I at a dead end? Should I instead, just begin with $\nabla f(x)$ in place of $f(x)$ when I apply Taylor's theorem?