Differentiate sums and compositions

Differentiate sums, products, quotients and compositions

For real constant a, the source lists

$$\frac d{dx}(a)=0,\qquad \frac d{dx}[af(x)]=a\frac d{dx}f(x),\qquad \frac d{dx}[f(x)\pm g(x)]=\frac d{dx}f(x)\pm\frac d{dx}g(x).$$

Example 24 rewrites the reciprocal and radical as powers. For differentiable u,v the product rule is

$$\frac{dy}{dx}=\frac d{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}.$$

Example 25 retains the explicitly requested product-rule method. For a nonzero denominator v, the quotient rule is

$$\frac{dy}{dx}=\frac d{dx}\left(\frac uv\right) =\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}.$$

Example 26 keeps the numerator subtraction order and the full denominator square. For y a function of u and u a function of x, the chain rule is

$$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.$$

Example 27 applies it to a square root, distinguishing the real function’s endpoint from the open interval where its derivative is finite.

Retain the elementary derivative table

The source’s five basic/general pairs are retained below. Trigonometric derivatives use radians. The logarithm row requires x>0 in the basic form and f(x)>0 in the general form; tangent rows exclude their poles, and each inner function must be differentiable at the input.

Example 28 retains its exponential, product-with-sine and logarithm asks. Continue with Tutorial Question 16 and Tutorial Question 17.