Differentiate general exponentials

6. (a) Given that \(y = 2 ^ { x }\), and using the result \(2 ^ { x } = \mathrm { e } ^ { x \ln 2 }\), or otherwise, show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ^ { x } \ln 2\).
(b) Find the gradient of the curve with equation \(y = 2 ^ { \left( x ^ { 2 } \right) }\) at the point with coordinates \(( 2,16 )\).

12 A curve has equation \(y = a ^ { 3 x ^ { 2 } }\), where \(a\) is a constant greater than 1 .

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x a ^ { 3 x ^ { 2 } } \ln a\).
2. The tangent at the point \(\left( 1 , a ^ { 3 } \right)\) passes through the point \(\left( \frac { 1 } { 2 } , 0 \right)\). Find the value of \(a\), giving your answer in an exact form.
3. By considering \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) show that the curve is convex for all values of \(x\). \section*{OCR} \section*{Oxford Cambridge and RSA}

7. (a) Prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( a ^ { x } \right) = a ^ { x } \ln a .$$ A curve has the equation \(y = 4 ^ { x } - 2 ^ { x - 1 } + 1\).
(b) Show that the tangent to the curve at the point where it crosses the \(y\)-axis has the equation $$3 x \ln 2 - 2 y + 3 = 0 .$$ (c) Find the exact coordinates of the stationary point of the curve.
7. continued

3. (a) Given that \(a ^ { x } = \mathrm { e } ^ { k x }\), where \(a\) and \(k\) are constants, \(a > 0\) and \(x \in \mathbb { R }\), prove that \(k = \ln a\).
(b) Hence, using the derivative of \(\mathrm { e } ^ { k x }\), prove that when \(y = 2 ^ { x }\), $$\frac { \mathrm { d } y } { \mathrm { dx } } = 2 ^ { x } \ln 2 .$$ (c) Hence deduce that the gradient of the curve with equation \(y = 2 ^ { x }\) at the point \(( 2,4 )\) is \(\ln 16\).

4 A curve has equation \(y = x ^ { 4 } + 2 ^ { x }\) Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)