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I am trying to find the derivatives of a sine wave function $$f(x,A,F,H,V) = A\sin(F(x-H))+V$$ with respect to each variable, where $A$ is amplitude, $F$ is frequency, $H$ is horizontal shift, and $V$ is vertical shift. So far I believe I have correctly found the derivatives with respect to amplitude and vertical shift.

$\frac{\partial f}{\partial A} = \sin(F(x-H))$

$\frac{\partial f}{\partial V} = 1$

However I am having trouble finding the derivative with respect to frequency and horizontal shift. Kindly let me know how I should go about this and let me know if I have made any errors in derivatives with respect to amplitude and vertical shift.

Alex Provost
  • 20,991
abnj77
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1 Answers1

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I'll show you how to do the partial derivative with respect to $F$ and leave $H$ as an exercise.

$$\begin{align*} \frac\partial{\partial F} \big(A\sin(F(x-H))+V\big) &= A\frac\partial{\partial F} \sin(F(x-H)) + 0 \quad \text{(linearity)}\\ &= A\cos(F(x-H))\frac\partial{\partial F}(F(x-H)) \quad \text{(chain rule)}\\ &=A\cos(F(x-H))(x-H). \end{align*}$$

Alex Provost
  • 20,991