Let where is a constant

MATHEMATICS · Subjective · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let f(x)=sin⁡(ax)f(x)=\sin(ax) where a (≠0)a\,(\neq 0) is a constant. (i) Write the definition of f′(x)f'(x) as a limit involving an increment hh. (ii) Simplify the numerator sin⁡(a(x+h))−sin⁡(ax)\sin\bigl(a(x+h)\bigr)-\sin(ax) by using a suitable trigonometric identity. (iii) Hence evaluate the limit to obtain ddx(sin⁡(ax))\displaystyle \frac{d}{dx}\bigl(\sin(ax)\bigr). (iv) Using the result of part (iii), find the slope of the tangent to the curve y=sin⁡(5x)y=\sin(5x) at the point where x=π6x=\frac{\pi}{6}.

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